#7086921

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,27)(22,24)(23,28)(25,32)(30,43)(31,40)(33,38)(34,41)(35,42)(36,37)(39,46)(44,48)(45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(21,22)(25,37)(27,28)(29,40)(30,44)(31,36)(34,35)(38,41)(45,46)(47,48);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,37)(26,39)(27,40)(28,29)(33,44)(34,45)(35,46)(38,47)(41,48)(42,43);(11,16)(13,15)(17,20,23,35)(18,26,28,42)(19,30,32,46)(21,34,22,33)(24,38,27,41)(25,39,29,43)(31,44,36,45)(37,47,40,48);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,31,32,36)(20,34,35,33)(25,37,29,40)(26,41,42,38)(30,44,46,45)(39,47,43,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,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, 6994117);