#6543345

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,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, 6450541);