#6490045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(11,14)(12,15)(13,16)(17,18,22,28)(19,29,32,24)(20,26,34,42)(21,27,23,25)(30,43,44,38)(31,37,33,39)(35,40,36,41)(45,48,46,47);(2,5)(6,8)(9,11,14,10)(12,15,13,16)(17,19,22,32)(18,24,28,29)(20,30,34,44)(21,31,23,33)(25,37,27,39)(26,38,42,43)(35,45,36,46)(40,47,41,48);(9,12,14,13)(10,15,11,16)(18,25)(20,34)(21,23)(24,37)(26,41)(27,28)(29,39)(30,44)(31,33)(38,48)(40,42)(43,47);(9,12,14,13)(10,15,11,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,38)(25,40)(27,41)(28,42)(29,43)(31,46)(32,44)(33,45)(37,47)(39,48);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,27,28,25)(19,31,32,33)(20,35,34,36)(24,39,29,37)(26,40,42,41)(30,45,44,46)(38,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6397241);