#6509254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,33)(21,26)(22,36)(23,28)(25,34)(27,37)(29,43)(30,39)(31,40)(32,47)(35,42)(38,44)(41,46)(45,48);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,38)(26,39)(27,41)(28,40)(33,43)(34,44)(35,45)(36,47)(37,46)(42,48);(10,12)(11,16)(17,20,23,21)(18,25,28,42)(19,29,31,30)(22,35,37,34)(24,38,40,48)(26,36,33,27)(32,45,46,44)(39,47,43,41);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,21,23,20)(18,26,28,33)(19,30,31,29)(22,35,37,34)(24,39,40,43)(25,36,42,27)(32,45,46,44)(38,47,48,41);(11,16)(14,15)(19,31)(20,21)(24,40)(25,42)(26,33)(32,46)(34,35)(41,47)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

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, 6416450);