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 | |
4125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,23,21)(19,26,24,20)(22,29,31,25)(27,28,32,30)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(33,35,39,43)(34,36,37,40)(38,42,47,48)(41,44,45,46);(2,6)(3,9)(8,11)(10,12)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,36,39,40)(34,35,37,43)(38,46,47,44)(41,48,45,42);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,19)(21,24)(22,28)(23,26)(25,27)(29,32)(30,31)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,46,48,44);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28)(33,35)(34,40)(36,37)(38,42)(39,43)(41,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837488);