#93609

order	512
nilpotency class	4
isoclinism	isoclinic class of #93517
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

194 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,c^b=cg,d^b=dh,g^b=gi,h^b=hi

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, 805);