#476296

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1223 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383492);