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 | |
4180 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)(3,7)(4,9)(5,10)(6,11)(8,13)(12,14)(15,16)(20,28)(24,31)(26,32)(29,30)(33,34,39,42)(35,43,46,38)(36,37,41,40)(44,47,48,45);(1,3,6,13)(2,7,11,8)(4,10,14,16)(5,12,15,9)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,39,46)(34,38,42,43)(36,44,41,48)(37,45,40,47);(1,3)(2,8)(4,10)(5,12)(6,13)(7,11)(9,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,41)(34,40,42,37)(35,44,46,48)(38,47,43,45);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,45)(47,48);(1,4,6,14)(2,9,11,12)(3,10,13,16)(5,8,15,7)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,39,40)(34,41,42,36)(35,45,46,47)(38,48,43,44);(1,5)(2,10)(3,12)(4,8)(6,15)(7,14)(9,13)(11,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,38,39,43)(34,35,42,46)(36,47,41,45)(37,48,40,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=h,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, 8337962);