#499010

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1066 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)(7,13)(8,23)(9,15)(10,30)(11,34)(12,17)(14,26)(16,32)(18,46)(19,36)(20,50)(21,38)(22,39)(24,55)(25,56)(27,44)(28,57)(29,45)(31,41)(33,60)(35,47)(37,52)(40,51)(42,58)(43,59)(48,67)(49,66)(53,65)(54,64)(61,63)(62,68);(1,2)(3,4)(5,7)(6,13)(8,24)(9,19)(10,20)(11,21)(12,22)(14,40)(15,36)(16,37)(17,39)(18,27)(23,55)(25,48)(26,51)(28,54)(29,49)(30,50)(31,61)(32,52)(33,53)(34,38)(35,42)(41,63)(43,62)(44,46)(45,66)(47,58)(56,67)(57,64)(59,68)(60,65);(1,3,4,2)(5,8)(6,14)(7,18)(9,25)(10,31)(11,27)(12,28)(13,35)(15,41)(16,44)(17,43)(19,24)(20,51)(21,54)(22,48)(23,30)(26,32)(29,59)(33,58)(34,42)(36,40)(37,64)(38,62)(39,63)(45,56)(46,50)(47,52)(49,61)(53,68)(55,66)(57,60)(65,67);(5,9,12,11)(6,15,17,34)(7,19,22,21)(8,25,28,27)(10,29,33,32)(13,36,39,38)(14,41,43,42)(16,30,45,60)(18,24,48,54)(20,49,53,52)(23,56,57,44)(26,31,59,58)(35,40,63,62)(37,50,66,65)(46,55,67,64)(47,51,61,68)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406206);