#93447

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#333
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1296
prime radical	C₂³	#343
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

210 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=h,e^2=fgi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi

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, 643);