#6567450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 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)(9,10,14,13)(11,16,12,15)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,32)(25,36)(29,41)(30,45)(31,40)(34,39)(35,43)(37,42)(44,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,40,38,45)(20,34,36,47,35,29,37,46)(25,41,33,48,42,39,43,44);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,22,36,23,35,21,37)(18,25,26,33,27,42,38,43)(19,29,32,46,30,34,31,47)(24,39,28,44,40,41,45,48);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(19,30)(20,35)(24,40)(25,42)(28,45)(31,32)(33,43)(36,37);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,26,27,38)(19,31,30,32)(20,36,35,37)(24,28,40,45)(25,43,42,33)(29,46,34,47)(39,48,41,44)

Polycyclic presentation:

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

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, 6474646);