#93568

order	512
nilpotency class	4
isoclinism	isoclinic class of #93537
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1294
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×#97	C₂²
prime radical	C₂⁴	#103
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 764);