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 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34,38,40)(35,39,43,45)(36,41,46,44)(37,47,48,42);(2,7)(4,8)(10,15)(12,13)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35,38,43)(34,39,40,45)(36,42,46,47)(37,44,48,41);(18,24)(20,28)(23,30)(25,31)(34,40)(35,43)(41,44)(42,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,45)(44,46)(47,48);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,41)(35,42)(37,39)(38,46)(40,44)(43,47)(45,48);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,37)(34,42)(35,44)(36,45)(38,48)(39,46)(40,47)(41,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817249);