#72319762

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²⋊C₄×#102
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#576	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3656 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)(3,10)(4,11)(5,26,31,12)(6,13)(7,14)(8,16)(9,15)(17,44,27,35)(18,36)(19,37)(20,39)(21,38)(22,40)(23,41)(24,45,28,42)(25,43)(29,57,51,46)(30,58,54,47)(32,59,55,48)(33,49)(34,50)(52,63,56,61)(53,64,60,62);(1,3,6,18,23,4,22,19,9,21,33,25,8,20,7,34)(2,10,13,36,41,11,40,37,15,38,49,43,16,39,14,50)(5,27,29,56,55,28,54,60,31,17,51,52,32,24,30,53)(12,44,46,63,59,45,58,64,26,35,57,61,48,42,47,62);(1,4,23,3,9,20,8,21)(2,11,41,10,15,39,16,38)(5,28,55,27,31,24,32,17)(6,18,22,34,33,25,7,19)(12,45,59,44,26,42,48,35)(13,36,40,50,49,43,14,37)(29,56,54,53,51,52,30,60)(46,63,58,62,57,61,47,64);(1,5,9,31)(2,12,15,26)(3,17,21,27)(4,24,20,28)(6,29,33,51)(7,30,22,54)(8,32,23,55)(10,35,38,44)(11,42,39,45)(13,46,49,57)(14,47,40,58)(16,48,41,59)(18,52,25,56)(19,53,34,60)(36,61,43,63)(37,62,50,64)

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efij,d^2=j,e^2=ij,f^2=i,i^2=j,c^b=cf,d^a=dg,d^b=dj,d^c=dj,e^c=ei,f^b=fj,f^c=fi,g^a=gj,h^d=hj,i^c=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319811, #72320989, #72343978
maximal normal subgroup of	#72319811, #72320989, #72343978

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