#94568

order	512
nilpotency class	6
isoclinism	isoclinic class of #7529
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1408
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	#1845	C₂²
prime radical	C₂³	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1845 ◁ G
LECS	G ▷ #1845 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

169 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=gh,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cghi,d^b=deg,e^b=ehi,g^b=gi,d^c=dgh,e^d=ehi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1764);