#72257355

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#127
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂³	C₂×#392
prime radical	C₂³×#41	C₂³
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

4655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=e,c^2=i,d^2=gh,f^2=j,b^a=beg,c^b=cf,d^a=dg,d^b=dh,d^c=defi,f^b=fj,f^d=fj,h^c=hj,i^b=ij,i^d=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257425, #72264071
maximal normal subgroup of	#72257425, #72264071

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