#501039

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

972 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)(5,6,12,17,13,18,10,11)(7,19,23,14,24,36,22,40)(8,25,29,15,30,37,28,42)(9,31,35,16,34,33,32,43)(20,38,48,44,49,55,47,26)(21,39,52,45,51,62,50,59)(27,41,58,53,57,63,56,60)(46,64,67,61,66,54,65,68);(2,4)(5,7,13,24)(6,14,18,40)(8,26,30,44)(9,21,34,51)(10,23,12,22)(11,36,17,19)(15,20,42,49)(16,39,43,62)(25,48,37,47)(27,54,57,64)(28,38,29,55)(31,45,33,59)(32,52,35,50)(41,46,63,66)(53,65,60,67)(56,68,58,61);(5,8,13,30)(6,15,18,42)(7,20,24,49)(9,27,34,57)(10,29,12,28)(11,37,17,25)(14,38,40,55)(16,41,43,63)(19,44,36,26)(21,46,51,66)(22,48,23,47)(31,53,33,60)(32,58,35,56)(39,61,62,68)(45,64,59,54)(50,67,52,65);(2,4)(5,9)(6,16)(7,21)(8,27)(10,35)(11,33)(12,32)(13,34)(14,39)(15,41)(17,31)(18,43)(19,45)(20,46)(22,52)(23,50)(24,51)(25,53)(26,54)(28,58)(29,56)(30,57)(36,59)(37,60)(38,61)(40,62)(42,63)(44,64)(47,67)(48,65)(49,66)(55,68)

Polycyclic presentation:

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

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, 408235);