#72303118

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#11087
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
socle	C₂	C₂×#11087
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5341 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=j,d^2=efij,b^a=bh,d^a=de,d^b=dg,d^c=dj,e^a=ej,e^b=ej,e^d=ej,f^d=fj,g^a=gi,g^b=gi,h^d=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303123, #72303324, #72303420
maximal normal subgroup of	#72303123, #72303324, #72303420

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.6);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);