#135005

order	512
nilpotency class	4
isoclinism	isoclinic class of #134965
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,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, 42201);