#72312753

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, socle	C₂²	#23900
commutator	C₂×#41	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	C₂×#33570	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

4737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,13)(8,14)(9,12)(10,15)(11,16)(17,24)(18,28)(19,29)(20,31)(21,25)(22,26)(23,30)(27,33)(32,36)(34,35);(7,17,19,22)(13,26,29,24)(18,32,34,33)(27,35,36,28);(6,12)(7,18,19,34)(8,20)(9,10)(13,27,29,36)(14,30)(15,16)(17,32,22,33)(24,28,26,35)(25,31);(1,3,4,2)(5,7)(6,13)(8,18)(9,17)(10,22)(11,19)(12,24)(14,28)(15,26)(16,29)(20,34)(21,33)(23,32)(25,27)(30,36)(31,35)

Polycyclic presentation:

2^10|a^2=e,b^2=i,c^2=i,d^2=e,g^2=j,h^2=j,b^a=bi,c^a=cg,d^b=dh,d^c=df,f^b=fi,f^c=fj,g^a=gj,g^c=gj,h^a=hj,h^c=hij,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312768, #72312795, #72312884
maximal normal subgroup of	#72312768, #72312795, #72312884

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