#7010323

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

1354 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)(17,18,22,27,21,26,35,24)(19,28,31,37,30,23,36,39)(20,32,34,40,33,25,41,42)(29,38,45,48,44,43,46,47);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,37)(25,38)(26,28)(27,39)(32,43)(33,44)(34,45)(35,36)(40,47)(41,46)(42,48);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(18,24)(22,35)(23,37)(25,40)(26,27)(28,39)(31,36)(32,42)(34,41)(38,47)(43,48)(45,46);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20,21,33)(18,25,26,32)(19,29,30,44)(22,34,35,41)(23,38,28,43)(24,40,27,42)(31,45,36,46)(37,47,39,48);(10,14)(13,15)(19,30)(20,33)(23,28)(25,32)(31,36)(34,41)(37,39)(40,42)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=f,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, 6917519);