#494413

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

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	C₂×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1272 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:

(2,4)(5,6,10,16,11,17,33,47,12,18,34,48,9,15,20,36)(7,19,24,50,23,38,51,45,22,43,31,59,29,13,35,39)(8,25,27,58,26,42,52,46,28,44,32,60,30,14,40,41)(21,37,55,64,56,49,61,66,53,62,57,65,54,63,68,67);(1,2,3,4)(5,7,20,51,9,29,34,24,12,22,33,35,11,23,10,31)(6,13,36,50,15,43,48,39,18,38,47,59,17,19,16,45)(8,21,52,68,30,54,27,57,28,53,40,61,26,56,32,55)(14,37,58,67,44,63,41,65,42,62,60,66,25,49,46,64);(5,8)(6,14)(7,21)(9,30)(10,32)(11,26)(12,28)(13,37)(15,44)(16,46)(17,25)(18,42)(19,49)(20,52)(22,53)(23,56)(24,57)(27,34)(29,54)(31,55)(33,40)(35,61)(36,58)(38,62)(39,65)(41,48)(43,63)(45,64)(47,60)(50,67)(51,68)(59,66);(7,22)(13,38)(19,43)(21,53)(23,29)(24,31)(35,51)(37,62)(39,45)(49,63)(50,59)(54,56)(55,57)(61,68)(64,65)(66,67)

Polycyclic presentation:

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

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, 401609);