#6483045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 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)(11,15)(12,16)(13,14)(17,18,24,30)(19,31,36,44)(20,37,38,43)(21,27,32,42)(22,29,23,28)(25,35,45,34)(26,33,46,39)(40,48,41,47);(2,5)(6,8)(10,14)(11,12)(17,19,22,34,24,36,23,35)(18,25,28,31,30,45,29,44)(20,32,33,41,38,21,39,40)(26,42,43,48,46,27,37,47);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,33,24,38,23,39)(18,26,28,43,30,46,29,37)(19,32,34,41,36,21,35,40)(25,42,31,48,45,27,44,47);(9,12,13,11)(10,16,14,15)(17,21,24,32)(18,27,30,42)(19,33,36,39)(20,34,38,35)(22,40,23,41)(25,43,45,37)(26,31,46,44)(28,47,29,48);(10,14)(15,16)(20,38)(21,32)(26,46)(27,42)(33,39)(37,43)(40,41)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei

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, 6390241);