#7083466

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1366 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)(19,31)(20,36)(21,27)(22,29)(23,28)(24,30)(25,34)(26,32)(33,45)(35,44)(37,43)(38,48)(39,46)(40,47)(41,42);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,34,24,35,22,33)(18,25,29,31,30,45,28,44)(20,21,37,38,32,39,40,41)(26,42,43,46,36,48,47,27);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,32)(18,26,30,36)(19,21,35,39)(22,40,23,37)(25,42,45,48)(27,31,46,44)(28,47,29,43)(33,41,34,38);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,32)(20,35)(22,38)(23,41)(24,39)(25,43)(26,31)(28,42)(29,48)(30,46)(33,40)(34,37)(36,44)(45,47);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,32,40)(21,38,39,41)(25,44,45,31)(26,43,36,47)(27,42,46,48)

Polycyclic presentation:

2^9|b^2=fgi,c^2=gh,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6990662);