#494230

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

974 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)(5,6,10,16,11,15,30,40,12,17,31,41,9,27,29,53)(7,13,21,35,22,34,50,60,19,33,46,59,20,47,49,65)(8,23,25,51,24,14,36,37,26,38,52,61,28,39,54,62)(18,42,45,63,44,32,55,57,43,58,64,68,48,56,66,67);(2,4)(5,7,10,21,11,22,30,50,12,19,31,46,9,20,29,49)(6,13,16,35,15,34,40,60,17,33,41,59,27,47,53,65)(8,18,25,45,24,44,36,55,26,43,52,64,28,48,54,66)(14,32,37,57,38,58,61,68,39,56,62,67,23,42,51,63);(5,8)(6,14)(7,18)(9,28)(10,25)(11,24)(12,26)(13,32)(15,38)(16,37)(17,39)(19,43)(20,48)(21,45)(22,44)(23,27)(29,54)(30,36)(31,52)(33,56)(34,58)(35,57)(40,61)(41,62)(42,47)(46,64)(49,66)(50,55)(51,53)(59,67)(60,68)(63,65);(7,19)(13,33)(18,43)(20,22)(21,46)(32,56)(34,47)(35,59)(42,58)(44,48)(45,64)(49,50)(55,66)(57,67)(60,65)(63,68)

Polycyclic presentation:

2^9|a^2=g,b^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,f^c=fi

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, 401426);