#509158

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×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 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)(5,6,10,17,11,18,40,55,13,20,42,57,12,19,41,56)(7,21,24,47,25,29,62,49,27,60,63,48,26,53,38,14)(8,15,30,51,43,58,68,67,32,52,66,61,31,23,46,34)(9,33,35,59,44,65,50,28,37,64,45,22,36,54,39,16);(5,7,13,27)(6,14,20,49)(8,28,32,16)(9,34,37,67)(10,38,42,62)(11,26,12,25)(15,50,52,39)(17,53,57,29)(18,48,19,47)(21,55,60,56)(22,43,59,31)(23,45,58,35)(24,40,63,41)(30,65,66,54)(33,46,64,68)(36,51,44,61);(2,4)(5,8)(6,15)(7,22)(9,29)(10,30)(11,43)(12,31)(13,32)(14,45)(16,25)(17,51)(18,58)(19,23)(20,52)(21,36)(24,54)(26,28)(27,59)(33,62)(34,56)(35,49)(37,53)(38,64)(39,47)(40,68)(41,46)(42,66)(44,60)(48,50)(55,67)(57,61)(63,65);(2,4)(5,9,13,37)(6,16,20,28)(7,23,27,58)(8,29,32,53)(10,39,42,50)(11,36,12,44)(14,46,49,68)(15,25,52,26)(17,54,57,65)(18,22,19,59)(21,31,60,43)(24,61,63,51)(30,47,66,48)(33,55,64,56)(34,62,67,38)(35,40,45,41)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=i,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, 416354);