#509159

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2192	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

780 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)(5,6,13,37,10,18,35,46,12,20,9,17,11,19,34,54)(7,21,56,44,25,59,32,14,27,60,24,43,26,53,38,45)(8,15,39,51,36,55,33,16,31,49,29,47,30,48,40,52)(22,57,23,42,64,68,65,41,62,58,61,67,28,50,63,66);(5,7)(6,14)(8,28)(9,32)(10,26)(11,25)(12,27)(13,38)(15,42)(16,50)(17,53)(18,44)(19,43)(20,45)(21,46)(22,30)(23,29)(24,35)(31,64)(33,65)(34,56)(36,62)(37,59)(39,61)(40,63)(41,48)(47,58)(49,67)(51,57)(52,68)(54,60)(55,66);(2,4)(5,8)(6,15)(7,22)(9,29)(10,36)(11,30)(12,31)(13,39)(14,41)(16,46)(17,47)(18,55)(19,48)(20,49)(21,57)(23,56)(24,61)(25,64)(26,28)(27,62)(32,65)(33,35)(34,40)(37,51)(38,63)(42,44)(43,67)(45,66)(50,53)(52,54)(58,60)(59,68);(2,4)(5,8)(6,16)(7,23)(9,33)(10,30)(11,36)(12,31)(13,40)(14,42)(15,46)(17,48)(18,51)(19,47)(20,52)(21,58)(22,56)(24,62)(25,63)(26,65)(27,61)(28,32)(29,35)(34,39)(37,55)(38,64)(41,44)(43,66)(45,67)(49,54)(50,53)(57,60)(59,68)

Polycyclic presentation:

2^9|a^2=ei,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,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, 416355);