#503155

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1112 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:

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

Polycyclic presentation:

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

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, 410351);