#6546322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453518);