#7622983

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,c^b=ci,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530179);