#135092

order	512
nilpotency class	4
isoclinism	isoclinic class of #134965
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,e^c=ehi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42288);