#494412

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

1280 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)(5,6,10,17,11,18,33,48,12,15,34,43,9,16,20,36)(7,19,23,50,22,38,51,46,24,44,31,59,29,13,35,39)(8,25,27,58,26,41,52,47,28,45,32,60,30,14,40,42)(21,37,54,64,55,49,61,66,56,62,57,65,53,63,68,67);(1,2,3,4)(5,7,20,51,9,29,34,23,12,24,33,35,11,22,10,31)(6,13,36,50,16,44,43,39,15,38,48,59,18,19,17,46)(8,21,52,68,30,53,27,57,28,56,40,61,26,55,32,54)(14,37,58,67,45,63,42,65,41,62,60,66,25,49,47,64);(5,8)(6,14)(7,21)(9,30)(10,32)(11,26)(12,28)(13,37)(15,41)(16,45)(17,47)(18,25)(19,49)(20,52)(22,55)(23,57)(24,56)(27,34)(29,53)(31,54)(33,40)(35,61)(36,58)(38,62)(39,65)(42,43)(44,63)(46,64)(48,60)(50,67)(51,68)(59,66);(6,15)(13,38)(14,41)(16,18)(17,43)(19,44)(25,45)(36,48)(37,62)(39,46)(42,47)(49,63)(50,59)(58,60)(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^a=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, 401608);