#93456

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#340
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 652);