#503164

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
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

1125 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)(10,17)(11,16)(12,35)(13,21)(14,28)(18,33)(20,51)(22,39)(23,38)(24,40)(26,60)(27,58)(29,59)(30,64)(31,46)(32,45)(34,50)(36,48)(37,65)(41,66)(42,62)(43,63)(44,52)(47,53)(49,55)(54,67)(56,68)(57,61);(1,2,4,3)(5,7,9,21)(6,13,15,19)(8,26,27,61)(10,23,31,53)(11,34,32,22)(12,36,33,24)(14,41,42,68)(16,39,45,50)(17,47,46,38)(18,48,35,40)(20,52,54,29)(25,57,58,60)(28,56,62,66)(30,65,63,55)(37,64,49,43)(44,51,59,67);(5,8,11,29,12,30,31,62,9,27,32,52,33,63,10,28)(6,14,17,43,18,44,45,58,15,42,46,64,35,59,16,25)(7,20,53,61,24,56,34,65,21,54,23,26,36,66,22,55)(13,37,50,68,40,57,47,51,19,49,39,41,48,60,38,67);(1,3,4,2)(6,15)(13,19)(14,42)(16,45)(17,46)(18,35)(25,58)(37,49)(38,47)(39,50)(40,48)(41,68)(43,64)(44,59)(51,67)(57,60)

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=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,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, 410360);