#93318

order	512
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄×#97	C₂²
socle	C₂²	#1288
prime radical	C₂³	#339
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514);