#6447030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,26)(21,36)(22,28)(23,27)(25,35)(29,39)(30,46)(31,41)(32,40)(33,45)(34,43)(37,44)(38,47)(42,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,24,28,41)(20,33,35,48)(21,32,23,30)(25,42,26,45)(27,46,36,40)(29,37,47,34)(38,43,39,44);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,28)(27,43)(30,33)(31,47)(32,48)(36,44)(39,41)(40,45)(42,46);(1,4,5,3)(2,8,6,7)(17,20)(18,26)(19,29)(21,34)(22,35)(23,37)(24,39)(25,28)(27,44)(30,33)(31,47)(32,48)(36,43)(38,41)(40,42)(45,46);(17,21,22,23)(18,27,28,36)(19,30,31,32)(20,34,35,37)(24,40,41,46)(25,43,26,44)(29,33,47,48)(38,45,39,42)

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354226);