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 | |
4164 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,7)(4,8)(9,14)(12,15)(18,19)(23,25)(24,26)(30,31)(35,43)(36,46)(39,48)(40,45);(5,13)(9,14)(10,16)(12,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109601);