#6741569

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,21)(19,24,30,28)(20,32,34,25)(23,26,37,36)(27,45,47,39)(29,40,31,41)(33,48,46,42)(35,44,38,43);(1,3)(2,5)(4,8)(6,7)(17,19,22,30)(18,24,21,28)(20,33,34,46)(23,29,37,31)(25,42,32,48)(26,40,36,41)(27,38,47,35)(39,44,45,43);(2,6)(5,7)(10,14)(15,16)(17,20,22,34)(18,25,21,32)(19,27,30,47)(23,38,37,35)(24,39,28,45)(26,44,36,43)(29,33,31,46)(40,42,41,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,18)(19,28,30,24)(20,25,34,32)(23,36,37,26)(27,39,47,45)(29,41,31,40)(33,42,46,48)(35,43,38,44);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,29)(20,34)(23,37)(24,40)(25,32)(26,36)(27,46)(28,41)(30,31)(33,47)(39,48)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648765);