#509207

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^c=fh,h^c=hi,e^d=ei,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, 416403);