#509618

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416814);