#500588

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,g^c=gi,h^c=hi

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, 407784);