#151541

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₁₆	D₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=fh,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=egi,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=egi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58737);