#72306738

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#13083
commutator	C₂×#41	C₂³×C₄
Frattini, cosocle	#532	C₂⁴
socle	C₂	C₂×#13083
prime radical	C₂×#23994	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=g,g^2=ij,h^2=j,b^a=bj,c^b=ci,d^a=de,d^b=df,e^b=eij,f^a=fi,f^b=fh,f^c=fj,f^e=fj,g^d=ghj,g^f=gij,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306739, #72306997
maximal normal subgroup of	#72306739, #72306997

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