#509168

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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
prime radical	C₂×#2193	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

765 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:

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

Polycyclic presentation:

2^9|a^2=ei,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416364);