#94490

order	512
nilpotency class	5
isoclinism	isoclinic class of #94431
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂²×D₄	C₄²
Frattini, cosocle	#2297	C₂²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,e^2=fi,g^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=df,g^b=gh,d^c=di,e^c=ehi,g^c=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1686);