#72262512

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#346446	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

2912 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=h,e^2=h,b^a=bf,c^a=cj,c^b=ce,d^a=dj,d^b=dh,d^c=dgij,e^a=ei,e^b=eh,e^c=eh,e^d=ehj,f^a=fh,f^c=fij,f^e=fj,g^b=gh,i^b=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262514, #72262622, #72262692
maximal normal subgroup of	#72262514, #72262622, #72262692

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