#499913

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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 ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1354 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:

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

Polycyclic presentation:

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

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, 407109);