#6578347

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,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, 6485543);