#6485930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,28)(19,31,35,44)(20,36,38,42)(21,27,37,45)(22,30,23,29)(25,34,43,33)(26,32,46,39)(40,48,41,47);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,33,24,35,23,34)(18,25,29,31,28,43,30,44)(20,37,32,41,38,21,39,40)(26,45,42,48,46,27,36,47);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,32,24,38,23,39)(18,26,29,42,28,46,30,36)(19,21,33,40,35,37,34,41)(25,27,31,47,43,45,44,48);(9,11,13,12)(10,15,14,16)(17,21)(18,27)(19,32)(20,34)(22,40)(23,41)(24,37)(25,42)(26,44)(28,45)(29,47)(30,48)(31,46)(33,38)(35,39)(36,43);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(18,28)(20,38)(21,37)(25,43)(29,30)(31,44)(32,39)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

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, 6393126);