#135003

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,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, 42199);