#494509

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(2,4)(5,6)(7,8)(9,10,14,21,36,50,37,53,16,19,13,20,15,22,40,56)(11,23,27,45,41,54,34,17,25,44,26,57,28,51,38,46)(12,29,31,49,42,55,35,18,33,48,30,58,32,52,39,47)(24,43,62,68,66,72,64,59,60,70,61,67,63,69,65,71);(1,2,3,4)(9,11)(10,17)(12,24)(13,34)(14,38)(15,41)(16,25)(18,43)(19,46)(20,51)(21,54)(22,57)(23,53)(26,37)(27,40)(28,36)(29,59)(30,64)(31,65)(32,66)(33,60)(35,61)(39,62)(42,63)(44,56)(45,50)(47,70)(48,71)(49,72)(52,67)(55,68)(58,69);(6,8)(9,12,16,33)(10,18,19,47)(11,24,25,60)(13,35,14,39)(15,42,36,32)(17,43,46,70)(20,52,21,55)(22,58,50,49)(23,59,44,71)(26,64,27,65)(28,66,41,63)(29,56,48,53)(30,40,31,37)(34,61,38,62)(45,72,57,69)(51,67,54,68);(10,19)(11,25)(18,47)(20,21)(22,50)(24,60)(26,27)(28,41)(29,48)(34,38)(49,58)(52,55)(53,56)(61,62)(63,66)(64,65)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401705);