#6398480

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂³	C₂⁴×C₄
prime radical	#36692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fh,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6305676);