#501035

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×#3417	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

964 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)(5,6,11,17,13,18,10,12)(7,19,23,14,24,36,22,39)(8,25,29,42,30,41,28,15)(9,31,33,16,35,34,32,43)(20,46,49,26,50,37,48,44)(21,38,52,45,53,63,51,61)(27,56,58,54,59,40,57,60)(47,55,67,65,68,64,66,62);(2,4)(5,7,13,24)(6,14,18,39)(8,26,30,44)(9,21,35,53)(10,23,11,22)(12,36,17,19)(15,20,42,50)(16,38,43,63)(25,48,41,49)(27,55,59,64)(28,37,29,46)(31,45,34,61)(32,52,33,51)(40,47,56,68)(54,67,60,66)(57,65,58,62);(5,8,13,30)(6,15,18,42)(7,20,24,50)(9,27,35,59)(10,29,11,28)(12,25,17,41)(14,37,39,46)(16,40,43,56)(19,44,36,26)(21,47,53,68)(22,49,23,48)(31,60,34,54)(32,58,33,57)(38,62,63,65)(45,64,61,55)(51,67,52,66);(2,4)(5,9)(6,16)(7,21)(8,27)(10,33)(11,32)(12,34)(13,35)(14,38)(15,40)(17,31)(18,43)(19,45)(20,47)(22,52)(23,51)(24,53)(25,54)(26,55)(28,58)(29,57)(30,59)(36,61)(37,62)(39,63)(41,60)(42,56)(44,64)(46,65)(48,67)(49,66)(50,68)

Polycyclic presentation:

2^9|a^2=f,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, 408231);