#135004

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

294 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dghi,d^2=gi,e^2=hi,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, 42200);