#419803

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1536 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)(5,6,10,16,12,17,11,34)(7,18,21,37,23,45,22,13)(8,24,28,41,29,40,27,14)(9,15,31,43,33,44,32,59)(19,35,48,38,49,61,25,46)(20,42,30,58,51,66,50,36)(26,52,56,64,57,63,55,39)(47,60,67,62,54,68,53,65);(5,7,12,23)(6,13,17,37)(8,25,29,48)(9,30,33,50)(10,22,11,21)(14,38,41,46)(15,42,44,66)(16,45,34,18)(19,28,49,27)(20,32,51,31)(24,35,40,61)(26,54,57,47)(36,59,58,43)(39,60,64,68)(52,65,63,62)(53,56,67,55);(5,8)(6,14)(7,19)(9,26)(10,27)(11,28)(12,29)(13,35)(15,39)(16,40)(17,41)(18,46)(20,47)(21,25)(22,48)(23,49)(24,34)(30,53)(31,55)(32,56)(33,57)(36,60)(37,61)(38,45)(42,65)(43,63)(44,64)(50,67)(51,54)(52,59)(58,68)(62,66);(2,4)(5,9,11,32,12,33,10,31)(6,15,34,59,17,44,16,43)(7,20,22,50,23,51,21,30)(8,26,28,56,29,57,27,55)(13,36,45,66,37,58,18,42)(14,39,24,52,41,64,40,63)(19,47,48,67,49,54,25,53)(35,60,38,62,61,68,46,65)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 326999);