#501038

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×#3416	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

977 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)(5,6,9,14,12,18,10,16)(7,19,22,38,25,36,23,13)(8,26,27,42,30,39,28,15)(11,17,31,40,33,45,32,43)(20,37,24,46,50,34,49,44)(21,35,51,60,54,62,52,47)(29,55,56,65,58,63,57,41)(48,64,67,59,68,66,53,61);(2,4)(5,7,12,25)(6,13,18,38)(8,21,30,54)(9,23,10,22)(11,20,33,50)(14,36,16,19)(15,35,42,62)(17,44,45,46)(24,31,49,32)(26,47,39,60)(27,52,28,51)(29,48,58,68)(34,43,37,40)(41,64,65,66)(53,57,67,56)(55,61,63,59);(6,14)(7,20)(9,10)(11,32)(13,34)(15,39)(16,18)(19,46)(21,48)(22,49)(23,24)(25,50)(26,42)(27,28)(29,56)(31,33)(35,59)(36,44)(37,38)(40,43)(47,66)(51,53)(52,67)(54,68)(55,63)(57,58)(60,64)(61,62);(2,4)(5,8)(6,15)(7,21)(9,28)(10,27)(11,29)(12,30)(13,35)(14,39)(16,26)(17,41)(18,42)(19,47)(20,48)(22,52)(23,51)(24,53)(25,54)(31,57)(32,56)(33,58)(34,59)(36,60)(37,61)(38,62)(40,63)(43,55)(44,64)(45,65)(46,66)(49,67)(50,68)

Polycyclic presentation:

2^9|a^2=fi,b^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, 408234);