#512376

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

793 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)(5,6,12,18)(7,19,20,38)(8,25,28,58)(9,17,11,15)(10,30,33,42)(13,24,35,22)(14,39,41,27)(16,29,44,32)(21,50,48,68)(23,37,49,63)(26,59,34,43)(31,45,61,40)(36,64,62,54)(46,66,65,52)(47,53,60,55)(51,57,56,67);(5,7,9,22,12,20,11,24)(6,13,15,19,18,35,17,38)(8,21,26,51,28,48,34,56)(10,23,29,53,33,49,32,55)(14,36,40,46,41,62,45,65)(16,37,42,47,44,63,30,60)(25,57,43,50,58,67,59,68)(27,52,31,54,39,66,61,64);(1,2,3,4)(7,20)(13,35)(19,38)(21,48)(22,24)(23,49)(36,62)(37,63)(46,65)(47,60)(50,68)(51,56)(52,66)(53,55)(54,64)(57,67);(5,8)(6,14)(7,21)(9,26)(10,31)(11,34)(12,28)(13,36)(15,40)(16,43)(17,45)(18,41)(19,46)(20,48)(22,51)(23,54)(24,56)(25,30)(27,32)(29,39)(33,61)(35,62)(37,50)(38,65)(42,58)(44,59)(47,67)(49,64)(52,55)(53,66)(57,60)(63,68)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419572);