#6567220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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)(22,37)(23,27)(24,31)(25,35)(29,44)(30,39)(32,43)(34,41)(36,40)(38,45)(42,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,36,47,34,45,35,46)(25,38,41,48,40,44,33,42);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,22,35,23,34,21,36)(18,25,26,33,27,40,37,41)(19,29,31,46,32,45,30,47)(24,38,28,42,39,44,43,48);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(20,34)(25,40)(29,45)(33,41)(35,36)(38,44)(42,48)(46,47);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,43)(25,41,40,33)(29,46,45,47)(38,48,44,42)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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^c=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474416);