#93992

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂²	C₈×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	C₄×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

(9,10,11,16,12,17,20,28,13,18,21,29,22,30,33,38,14,15,19,25,23,26,31,35,24,27,32,36,34,37,39,40);(1,2,3,5,4,6,7,8)(10,15)(11,19)(17,26)(18,27)(20,31)(21,32)(30,37)(33,39)

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,c^a=ci,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1188);