#135086

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

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	Q₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 250 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,4,7,5,3,8,6)(9,10,14,21,16,20,15,37)(11,22,26,50,28,49,27,59)(12,29,31,45,32,18,36,44)(13,33,23,46,35,39,34,19)(17,38,42,57,41,56,48,25)(24,52,54,70,55,47,58,69)(30,62,63,61,64,65,51,43)(40,66,67,53,60,71,68,72);(9,11,23,38,15,27,13,25,16,28,34,56,14,26,35,57)(10,17,39,59,21,42,19,22,20,41,33,50,37,48,46,49)(12,24,51,66,31,54,30,53,32,55,63,71,36,58,64,72)(18,40,62,69,45,68,43,47,29,60,65,70,44,67,61,52);(1,3)(2,5)(4,6)(7,8)(9,12)(10,18)(11,24)(13,30)(14,36)(15,31)(16,32)(17,40)(19,43)(20,29)(21,45)(22,47)(23,51)(25,53)(26,58)(27,54)(28,55)(33,65)(34,63)(35,64)(37,44)(38,66)(39,62)(41,60)(42,68)(46,61)(48,67)(49,52)(50,70)(56,71)(57,72)(59,69)

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,d^2=gi,e^2=hi,f^2=h,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, 42282);