#500849

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

937 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408045);