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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,10,12,7)(4,13,14,8)(5,9,15,16)(18,21)(23,26)(24,29)(30,32)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,48)(44,47);(1,3,6,12)(2,7,11,10)(4,9,14,16)(5,8,15,13)(20,28)(24,29)(25,31)(30,32)(34,39)(36,41)(40,42)(43,47);(4,14)(5,15)(8,13)(9,16)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,34,38,39)(35,40,45,42)(36,37,41,46)(43,44,47,48);(1,4)(2,8)(3,9)(5,10)(6,14)(7,15)(11,13)(12,16)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,5)(2,9)(3,8)(4,10)(6,15)(7,14)(11,16)(12,13)(17,20,22,28)(18,24,21,29)(19,25,27,31)(23,30,26,32)(33,36,38,41)(34,37,39,46)(35,43,45,47)(40,44,42,48);(1,3)(2,10)(4,9)(5,8)(6,12)(7,11)(13,15)(14,16)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8718376);