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 | |
3968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,4)(5,13,15,9)(11,12,16,14)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34,40,37)(35,43,39,41)(36,45,38,42)(44,47,46,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19,23,31)(18,24,28,29)(20,30,21,22)(25,27,26,32)(33,35,40,39)(34,41,37,43)(36,44,38,46)(42,47,45,48);(2,8)(3,10)(9,13)(11,16)(17,20,23,21)(18,25,28,26)(19,30,31,22)(24,27,29,32)(33,36,40,38)(34,42,37,45)(35,44,39,46)(41,47,43,48);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,14)(15,16)(17,21,23,20)(18,26,28,25)(19,22,31,30)(24,32,29,27)(33,37,40,34)(35,43,39,41)(36,42,38,45)(44,47,46,48);(1,4)(2,3)(5,14)(6,7)(8,10)(9,16)(11,13)(12,15)(17,22)(18,27)(19,20)(21,31)(23,30)(24,26)(25,29)(28,32)(33,38,40,36)(34,42,37,45)(35,44,39,46)(41,48,43,47);(1,5,6,15)(2,9,8,13)(3,11,10,16)(4,12,7,14)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,39,40,35)(34,43,37,41)(36,46,38,44)(42,48,45,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9623370);