#6566801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(20,32)(21,26)(22,27)(23,36)(24,30)(25,33)(29,44)(31,40)(34,41)(35,39)(37,42)(38,46)(43,47)(45,48);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,22,30)(18,24,27,28)(20,29,34,45)(21,35,23,31)(25,38,42,43)(26,40,36,39)(32,48,41,44)(33,47,37,46);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,22,34)(18,25,27,42)(19,29,30,45)(21,33,23,37)(24,38,28,43)(26,41,36,32)(31,46,35,47)(39,48,40,44);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(19,30)(24,28)(29,45)(31,35)(38,43)(39,40)(44,48)(46,47);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,21,22,23)(18,26,27,36)(19,31,30,35)(20,33,34,37)(24,39,28,40)(25,41,42,32)(29,46,45,47)(38,48,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473997);