Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,23,22)(19,24,29,28)(20,30,25,21)(26,32,31,27)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(2,7)(4,14)(8,11)(10,16)(17,19,23,29)(18,24,22,28)(20,26,25,31)(21,27,30,32)(34,39)(35,40)(41,46)(42,47);(17,20)(18,21)(19,26)(22,30)(23,25)(24,27)(28,32)(29,31)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,21,23,30)(18,25,22,20)(19,27,29,32)(24,31,28,26);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,22,23,18)(19,28,29,24)(20,30,25,21)(26,32,31,27)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8411933);