#499021

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1050 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)(5,6,12,18)(7,13,22,39)(8,23,25,58)(9,29,34,47)(10,17,11,16)(14,40,42,26)(15,32,48,33)(19,50,51,68)(20,53,30,62)(21,49,35,38)(24,57,56,66)(27,46,28,59)(31,43,61,44)(36,64,65,52)(37,63,45,55)(41,54,67,60);(1,2)(3,4)(5,7)(6,13)(8,24)(9,30)(10,35)(11,21)(12,22)(14,41)(15,45)(16,49)(17,38)(18,39)(19,27)(20,34)(23,57)(25,56)(26,60)(28,51)(29,62)(31,52)(32,55)(33,63)(36,43)(37,48)(40,54)(42,67)(44,65)(46,50)(47,53)(58,66)(59,68)(61,64);(1,3,4,2)(5,8)(6,14)(7,19)(9,31)(10,28)(11,27)(12,25)(13,36)(15,46)(16,44)(17,43)(18,42)(20,54)(21,56)(22,51)(23,29)(24,35)(26,33)(30,60)(32,40)(34,61)(37,66)(38,67)(39,65)(41,49)(45,57)(47,58)(48,59)(50,53)(52,55)(62,68)(63,64);(8,25)(14,42)(19,51)(23,58)(24,56)(26,40)(27,28)(31,61)(36,65)(41,67)(43,44)(46,59)(50,68)(52,64)(54,60)(57,66)

Polycyclic presentation:

2^9|a^2=i,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fh,g^c=gi,h^c=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, 406217);