#7083372

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

1359 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,26)(21,27)(22,28)(23,39)(24,29)(25,33)(31,46)(32,43)(34,44)(35,42)(36,45)(37,41)(38,48)(40,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32,24,34,23,33)(18,25,39,44,29,43,28,30)(20,21,36,40,31,38,35,37)(26,41,42,48,46,47,45,27);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,31)(18,26,29,46)(19,21,34,38)(22,36,23,35)(25,41,43,47)(27,44,48,30)(28,45,39,42)(32,40,33,37);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,31)(20,34)(22,37)(23,40)(24,38)(25,42)(26,44)(28,41)(29,48)(30,46)(32,36)(33,35)(39,47)(43,45);(10,14)(13,15)(17,22,24,23)(18,28,29,39)(19,32,34,33)(20,35,31,36)(21,37,38,40)(25,30,43,44)(26,42,46,45)(27,41,48,47)

Polycyclic presentation:

2^9|b^2=fgi,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 6990568);