#6578212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485408);