#72286552

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#106
commutator	C₂×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂³
prime radical	C₂⁵	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=efij,c^2=d,d^2=h,f^2=i,g^2=j,i^2=j,b^a=be,c^b=cefgij,d^b=dj,f^c=fi,g^c=gj,i^c=ij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286557, #72286576, #72286734, …
maximal normal subgroup of	#72286557, #72286576, #72286734, …
maximal quotient of	#72287019

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