#6615687

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 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)(19,27)(20,31)(21,34)(22,25)(23,29)(24,32)(26,36)(28,42)(30,40)(33,39)(35,38)(37,48)(41,43)(44,45)(46,47);(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,23,25,38)(20,28,32,43)(21,35,36,29)(24,37,31,45)(26,40,34,27)(33,48,47,44)(39,42,46,41);(2,6)(7,8)(10,13)(11,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,31)(26,39)(27,41)(29,44)(30,43)(34,46)(35,48)(36,47)(38,45)(40,42);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,32)(24,31)(28,43)(33,47)(37,45)(39,46)(41,42)(44,48);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(18,25)(23,38)(24,31)(26,34)(27,40)(37,45)(39,46)(41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6522883);