#92868

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	C₂×C₈²	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

205 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=hi,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=ghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 64);