#93443

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=h,e^2=fg,h^2=i,b^a=bc,c^a=cf,e^a=efgi,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 639);