#501054

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1039 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)(5,6,9,14,28,41,57,54,11,16,30,43,10,15,29,42)(7,12,18,33,19,34,50,64,21,36,52,66,20,35,51,65)(8,22,24,44,31,60,59,39,26,55,58,38,25,40,27,13)(17,45,46,37,23,53,56,62,48,67,68,61,47,63,49,32);(2,4)(5,7)(6,12)(8,23)(9,18)(10,20)(11,21)(13,37)(14,33)(15,35)(16,36)(17,25)(19,28)(22,53)(24,56)(26,47)(27,46)(29,51)(30,52)(31,48)(32,38)(34,41)(39,61)(40,45)(42,65)(43,66)(44,62)(49,58)(50,57)(54,64)(55,63)(59,68)(60,67);(5,8,11,26)(6,13,16,39)(7,17,21,48)(9,27,30,59)(10,31,28,25)(12,32,36,62)(14,40,43,60)(15,44,41,38)(18,49,52,56)(19,47,20,23)(22,54,55,42)(24,57,58,29)(33,63,66,53)(34,61,35,37)(45,64,67,65)(46,50,68,51);(2,4)

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408250);