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 | |
4156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,26)(19,27,30,23)(20,24,31,29)(21,25,32,28)(33,34,40,38)(35,44,39,41)(36,43,37,42)(45,47,46,48);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,15)(14,16)(17,18)(19,27)(20,29)(21,25)(22,26)(23,30)(24,31)(28,32)(33,35,40,39)(34,41,38,44)(36,45,37,46)(42,47,43,48);(2,6)(4,13)(7,9)(10,14)(20,31)(21,32)(24,29)(25,28)(33,36,40,37)(34,42,38,43)(35,45,39,46)(41,47,44,48);(1,3)(2,7)(4,14)(5,11)(6,9)(8,16)(10,13)(12,15)(17,19,22,30)(18,23,26,27)(20,25,31,28)(21,29,32,24)(33,37,40,36)(34,43,38,42)(35,45,39,46)(41,47,44,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,20,22,31)(18,24,26,29)(19,28,30,25)(21,23,32,27)(33,38,40,34)(35,44,39,41)(36,42,37,43)(45,47,46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,16)(17,21,22,32)(18,25,26,28)(19,29,30,24)(20,23,31,27)(33,39,40,35)(34,44,38,41)(36,45,37,46)(42,47,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947572);