#7074456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6981652);