#6431690

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle, prime radical	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,27)(21,23)(24,28)(25,30)(26,32)(29,31)(33,34,38,42,37,39,43,46)(35,40,44,47,36,41,45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,25)(18,22,23,30)(20,26,28,31)(24,29,27,32)(34,39)(40,41)(42,46)(47,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(19,25)(24,27)(26,31)(33,35,43,45,37,36,38,44)(34,40,46,48,39,41,42,47);(3,11)(6,9)(10,16)(13,15)(17,20,21,28)(18,24,23,27)(19,26,25,31)(22,29,30,32)(34,39)(40,41)(42,46)(47,48);(2,8)(6,9)(7,14)(13,15)(33,36,38,45,37,35,43,44)(34,41,42,48,39,40,46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cg,d^b=dfi,e^c=ei

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, 6338886);