#6615023

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(3,8,4,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,26)(19,27,30,43)(20,24,33,40)(21,35,28,25)(23,36,38,31)(29,42,32,46)(34,48,44,41)(37,47,39,45);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,28,36,22,30,21,31)(18,23,35,43,26,38,25,27)(20,32,44,45,33,29,34,47)(24,39,48,42,40,37,41,46);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,20,22,33)(18,24,26,40)(19,29,30,32)(21,34,28,44)(23,37,38,39)(25,41,35,48)(27,42,43,46)(31,45,36,47);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(20,33)(24,40)(29,32)(34,44)(37,39)(41,48)(42,46)(45,47);(11,16)(13,14)(19,30)(23,38)(27,43)(29,32)(31,36)(37,39)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,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, 6522219);