#509104

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416300);