#6578343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6485539);