#509615

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	#33612	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=eh,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, 416811);