#6543369

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

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,d^a=di,e^a=ei,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, 6450565);