#135076

order	512
nilpotency class	4
isoclinism	isoclinic class of #134945
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,22,25,39)(12,27,31,37)(13,28)(14,20)(15,35)(16,21)(17,18,30,36)(19,33)(23,34)(24,29,26,40)(32,38);(9,11,13,18,15,26,33,37,16,25,34,36,14,24,32,27)(10,17,19,22,20,12,28,29,21,30,38,39,35,31,23,40);(1,3)(2,5)(4,6)(7,8)(9,12)(10,18)(11,23)(13,29)(14,17)(15,30)(16,31)(19,26)(20,37)(21,36)(22,32)(24,38)(25,28)(27,35)(33,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42272);