#98895

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#380
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

366 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,c^2=h,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=egh,f^c=fg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6091);