#512189

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

583 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi

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, 419385);