#94458

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator, prime radical	C₂²×D₄	C₄²
Frattini, cosocle	C₄×#124	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₄×#124 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 224 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=fi,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=egi,g^a=gi,h^a=hi,c^b=cg,d^b=df,g^b=gh,h^b=hi,e^c=ehi,g^c=gi,g^e=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, 1654);