#145041

order	512
nilpotency class	4
isoclinism	isoclinic class of #145026
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

315 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fh,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52237);