#93991

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₁₆	D₄
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	#346
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1187);