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 | |
4022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18)(19,24)(20,31)(21,25)(22,26)(23,27)(28,29)(30,32)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19,23,30)(18,24,27,32)(20,21,26,28)(22,29,31,25)(34,38)(39,41)(40,44)(46,47);(17,20,23,26)(18,22,27,31)(19,28,30,21)(24,25,32,29)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,19)(18,24)(20,28)(21,26)(22,25)(23,30)(27,32)(29,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,21,23,28)(18,25,27,29)(19,26,30,20)(22,32,31,24);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22)(18,26)(19,29)(20,27)(21,32)(23,31)(24,28)(25,30)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=gh,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, 8343886);