#502191

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×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

727 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)(5,6,10,17)(7,19,23,14)(8,24,26,59)(9,13,30,36)(11,18,12,33)(15,42,43,25)(16,31,37,28)(20,41,32,40)(21,51,53,58)(22,55,56,35)(27,60,34,46)(29,44,61,45)(38,50,47,62)(39,52,48,66)(49,67,54,63)(57,64,68,65);(1,3,4,2)(6,13)(7,20)(9,28)(11,12)(14,38)(15,24)(16,18)(17,36)(19,47)(21,49)(22,56)(23,32)(25,29)(27,34)(30,31)(33,37)(35,40)(39,63)(41,55)(42,61)(43,59)(44,46)(45,60)(48,67)(51,65)(53,54)(57,68)(58,64);(2,3)(5,7,10,23)(6,14,17,19)(8,21,26,53)(9,22,30,56)(11,32,12,20)(13,35,36,55)(15,39,43,48)(16,38,37,47)(18,41,33,40)(24,58,59,51)(25,52,42,66)(27,54,34,49)(28,50,31,62)(29,57,61,68)(44,65,45,64)(46,63,60,67);(5,8)(6,15)(7,21)(9,29)(10,26)(11,27)(12,34)(13,24)(14,39)(16,46)(17,43)(18,44)(19,48)(20,49)(22,57)(23,53)(25,28)(30,61)(31,42)(32,54)(33,45)(35,58)(36,59)(37,60)(38,63)(40,64)(41,65)(47,67)(50,52)(51,55)(56,68)(62,66)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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, 409387);