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 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18)(19,21)(20,24)(22,26)(23,25)(27,29)(28,30)(31,32)(33,34,39,43)(35,40,42,37)(36,45,38,41)(44,48,46,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29)(33,35,39,42)(34,40,43,37)(36,46,38,44)(41,48,45,47);(2,6)(3,9)(8,11)(10,12)(17,18,22,26)(19,23,25,21)(20,24,30,28)(27,31,32,29)(33,36,39,38)(34,41,43,45)(35,44,42,46)(37,48,40,47);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32)(33,37,39,40)(34,42,43,35)(36,47,38,48)(41,44,45,46);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,29,23,31)(33,38,39,36)(34,41,43,45)(35,44,42,46)(37,47,40,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)(33,34)(35,40)(36,45)(37,42)(38,41)(39,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=i,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, 8837386);