#9040092

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947288);