#509099

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

Polycyclic presentation:

2^9|c^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, 416295);