#509213

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,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, 416409);