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 |
| prime radical | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,11,13,7)(4,14,12,8)(5,9,15,16)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,48,47);(1,3,6,13)(2,7,10,11)(4,5,12,15)(8,9,14,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,35,38,45)(34,39,42,43)(36,37,44,48)(40,41,46,47);(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,36,38,44)(34,40,42,46)(35,37,45,48)(39,41,43,47);(1,4)(2,8)(3,5)(6,12)(7,9)(10,14)(11,16)(13,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,34)(35,43)(36,40)(37,47)(38,42)(39,45)(41,48)(44,46);(1,2)(3,7)(4,8)(5,9)(6,10)(11,13)(12,14)(15,16)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,5)(2,9)(3,12)(4,13)(6,15)(7,14)(8,11)(10,16)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,37)(34,41)(35,44)(36,45)(38,48)(39,46)(40,43)(42,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,e^2=i,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, 8352840);