#72262779

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#347043	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

2917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,e^2=h,b^a=bg,c^a=ce,c^b=ch,d^a=dghj,d^b=dgi,d^c=df,e^a=eh,e^b=ehj,e^c=eh,e^d=ehi,f^a=fi,f^b=fh,f^e=fj,g^c=ghj,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262785, #72262800, #72262804
maximal normal subgroup of	#72262785, #72262800, #72262804
maximal quotient of	#72338913

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