#492864

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1572 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)(7,18)(8,14)(9,28)(10,31)(11,33)(12,17)(13,20)(15,30)(16,29)(19,41)(21,36)(22,52)(23,44)(24,37)(25,54)(26,56)(27,58)(32,40)(34,46)(35,51)(38,55)(39,53)(42,50)(43,49)(45,62)(47,59)(48,68)(57,61)(60,67)(63,66)(64,65);(5,7,10,21,29,49,30,50,12,23,9,20,11,22,32,51)(6,13,15,18,33,42,31,52,17,36,40,44,16,35,28,43)(8,19,26,47,53,65,55,66,24,45,25,46,27,48,57,67)(14,34,38,41,58,63,56,68,37,59,61,62,39,60,54,64);(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,24)(13,34)(15,38)(16,39)(17,37)(18,41)(20,46)(21,47)(22,48)(23,45)(28,54)(29,53)(30,55)(31,56)(32,57)(33,58)(35,60)(36,59)(40,61)(42,63)(43,64)(44,62)(49,65)(50,66)(51,67)(52,68);(8,24)(14,37)(19,45)(25,26)(27,53)(34,59)(38,61)(39,58)(41,62)(46,47)(48,65)(54,56)(55,57)(60,63)(64,68)(66,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=ghi,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400060);