#502187

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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
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

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409383);