#500719

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61086	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=e,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,d^c=di,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407915);