#499080

order	512
nilpotency class	4
isoclinism	isoclinic class of #498860
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1164 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gh,h^b=hi,f^c=fhi,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281854, #72282014, #72282220
maximal normal subgroup of	#72281854, #72282014, #72282220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406276);