#501073

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
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	#60459	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1054 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408269);