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 | |
4009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,18,23,26)(19,27,20,24)(21,22,25,32)(28,31,30,29)(33,34,39,42)(35,43,46,38)(36,37,41,40)(44,47,48,45);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(19,20)(24,27)(28,30)(29,31)(33,35,39,46)(34,38,42,43)(36,44,41,48)(37,45,40,47);(17,19,23,20)(18,24,26,27)(21,28,25,30)(22,29,32,31)(33,36,39,41)(34,40,42,37)(35,44,46,48)(38,47,43,45);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,20,23,19)(18,24,26,27)(21,30,25,28)(22,29,32,31)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,45)(47,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,21,23,25)(18,22,26,32)(19,28,20,30)(24,29,27,31)(33,37,39,40)(34,41,42,36)(35,45,46,47)(38,48,43,44);(1,6,7,2)(3,11,14,8)(4,9,13,15)(5,16,12,10)(17,22)(18,25)(19,29)(20,31)(21,26)(23,32)(24,30)(27,28)(33,38,39,43)(34,35,42,46)(36,47,41,45)(37,48,40,44)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353214);