#7094745

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,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=dhi,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, 7001941);