#135085

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,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, 42281);