#503170

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1115 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)(7,19)(8,25)(9,17)(10,33)(11,35)(12,18)(13,22)(14,30)(15,32)(16,34)(20,50)(21,39)(23,41)(24,40)(26,48)(27,47)(28,57)(29,43)(31,58)(36,49)(37,59)(38,55)(42,52)(44,62)(45,63)(46,51)(53,65)(54,68)(56,64)(60,67)(61,66);(5,7,10,22,12,24,34,23)(6,13,16,19,18,41,33,40)(8,26,29,59,31,61,64,60)(9,21,32,51,27,55,11,36)(14,42,44,50,45,65,57,68)(15,38,17,46,35,39,47,49)(20,28,52,63,54,62,53,30)(25,37,56,48,58,67,43,66);(1,2,3,4)(5,8,27,62,10,29,11,30,12,31,9,28,34,64,32,63)(6,14,15,43,16,44,17,25,18,45,35,56,33,57,47,58)(7,20,51,59,23,53,21,26,24,54,36,60,22,52,55,61)(13,37,49,50,40,66,39,42,41,67,46,68,19,48,38,65);(5,9,10,32,12,27,34,11)(6,15,16,17,18,35,33,47)(7,21,22,51,24,55,23,36)(8,28,29,63,31,62,64,30)(13,38,19,46,41,39,40,49)(14,43,44,25,45,56,57,58)(20,26,52,59,54,61,53,60)(37,65,48,68,67,42,66,50)

Polycyclic presentation:

2^9|b^2=e,c^2=fhi,d^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=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, 410366);