#501242

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×#3293	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1024 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)(9,32)(10,35)(11,18)(12,41)(13,19)(14,23)(16,21)(17,38)(22,44)(24,55)(25,63)(26,47)(27,52)(28,45)(29,65)(30,49)(31,51)(33,66)(34,67)(36,57)(37,56)(39,58)(40,50)(42,43)(46,61)(48,60)(53,62)(54,59)(64,68);(5,7,13,26)(6,14,19,46)(8,27,31,62)(9,22,34,60)(10,36,39,28)(11,25,12,24)(15,32,51,67)(16,44,54,48)(17,55,56,63)(18,57,41,45)(20,58,47,35)(21,40,59,30)(23,37,61,38)(29,33,64,42)(43,50,66,49)(52,68,53,65);(2,4)(5,8,13,31)(6,15,19,51)(7,21,26,59)(9,28,34,36)(10,29,39,64)(11,40,12,30)(14,43,46,66)(16,47,54,20)(17,48,56,44)(18,50,41,49)(22,38,60,37)(23,42,61,33)(24,62,25,27)(32,45,67,57)(35,65,58,68)(52,55,53,63);(5,9)(6,16)(7,22)(8,28)(10,27)(11,33)(12,42)(13,34)(14,44)(15,47)(17,43)(18,52)(19,54)(20,51)(21,38)(23,40)(24,29)(25,64)(26,60)(30,61)(31,36)(32,35)(37,59)(39,62)(41,53)(45,65)(46,48)(49,63)(50,55)(56,66)(57,68)(58,67)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 408438);