#93584

order	512
nilpotency class	4
isoclinism	isoclinic class of #93583
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂²
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 780);