#72307184

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24033
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#373663
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(3,14,18,43)(4,20)(5,26)(8,30,32,33)(10,37)(15,45,39,41)(16,38)(17,42,22,51)(19,47,28,44)(24,25)(27,40)(31,59,61,53)(35,62)(46,55,63,49)(50,64,57,60)(54,58);(1,2,7,13)(3,15)(4,21)(5,27,26,40)(6,11,29,12)(8,28)(9,24)(10,38,37,16)(14,41)(17,49,22,55)(18,39)(19,32)(20,36)(23,35)(25,34)(30,47)(31,60,61,64)(33,44)(42,46,51,63)(43,45)(48,54)(50,53,57,59)(52,62)(56,58);(1,3,16,47,29,43,5,28,7,18,38,44,6,14,26,19)(2,8,27,41,11,30,10,39,13,32,40,45,12,33,37,15)(4,22,56,64,24,42,52,57,20,17,48,60,25,51,23,50)(9,31,54,49,21,53,62,46,34,61,58,55,36,59,35,63);(1,4,6,25,7,20,29,24)(2,9,12,36,13,34,11,21)(3,17,14,42,18,22,43,51)(5,23,16,48,26,52,38,56)(8,31,33,59,32,61,30,53)(10,35,27,58,37,62,40,54)(15,46,45,49,39,63,41,55)(19,50,44,60,28,57,47,64)

Polycyclic presentation:

2^10|a^2=e,b^2=fj,c^2=fgi,d^2=i,g^2=i,h^2=j,i^2=j,b^a=bf,c^a=ch,c^b=cg,d^a=dfhi,d^b=dfh,d^c=dfhij,e^c=ej,f^c=fj,f^d=fj,g^a=gj,g^b=gi,g^c=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72320733, #72343925
maximal normal subgroup of	#72307301, #72320733, #72343925

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6*F.10);;
gap> SetPower(coll, 3, F.6*F.7*F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);