#501243

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1037 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)(7,20)(8,15)(9,30)(10,33)(11,18)(12,39)(13,19)(14,23)(16,31)(17,37)(21,54)(22,41)(24,50)(25,62)(26,56)(27,63)(28,49)(29,46)(32,47)(34,52)(35,48)(36,51)(38,53)(40,57)(42,60)(43,61)(44,64)(45,66)(55,59)(58,68)(65,67);(5,7,13,26)(6,14,19,43)(8,27,9,22)(10,34,38,60)(11,25,12,24)(15,44,30,55)(16,41,49,63)(17,50,51,62)(18,52,39,42)(20,53,56,33)(21,32,58,29)(23,36,61,37)(28,65,31,57)(35,59,66,64)(40,47,67,46)(45,54,48,68);(2,4)(5,8,13,9)(6,15,19,30)(7,21,26,58)(10,28,38,31)(11,32,12,29)(14,40,43,67)(16,33,49,53)(17,45,51,48)(18,47,39,46)(20,54,56,68)(22,25,27,24)(23,57,61,65)(34,64,60,59)(35,37,66,36)(41,62,63,50)(42,55,52,44);(5,9,13,8)(6,16,19,49)(7,22,26,27)(10,35,38,66)(11,32,12,29)(14,41,43,63)(15,33,30,53)(17,46,51,47)(18,48,39,45)(20,55,56,44)(21,25,58,24)(23,57,61,65)(28,37,31,36)(34,59,60,64)(40,62,67,50)(42,54,52,68)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^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, 408439);