#494229

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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 ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

982 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)(5,6,10,17,11,16,30,41,12,15,31,40,9,27,29,53)(7,13,20,35,21,34,49,60,22,33,50,59,19,46,48,65)(8,23,25,51,24,14,36,38,26,39,52,62,28,37,54,61)(18,42,44,63,43,32,55,57,45,58,64,68,47,56,66,67);(2,4)(5,7,10,20,11,21,30,49,12,22,31,50,9,19,29,48)(6,13,17,35,16,34,41,60,15,33,40,59,27,46,53,65)(8,18,25,44,24,43,36,55,26,45,52,64,28,47,54,66)(14,32,38,57,39,58,62,68,37,56,61,67,23,42,51,63);(5,8)(6,14)(7,18)(9,28)(10,25)(11,24)(12,26)(13,32)(15,37)(16,39)(17,38)(19,47)(20,44)(21,43)(22,45)(23,27)(29,54)(30,36)(31,52)(33,56)(34,58)(35,57)(40,61)(41,62)(42,46)(48,66)(49,55)(50,64)(51,53)(59,67)(60,68)(63,65);(6,15)(13,33)(14,37)(16,27)(17,40)(23,39)(32,56)(34,46)(35,59)(38,61)(41,53)(42,58)(51,62)(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,d^a=di,f^a=fi,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, 401425);