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 | |
4170 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,23,28,24)(20,21,30,32)(26,27,29,31)(33,34)(35,43)(36,46)(37,47)(38,42)(39,45)(40,44)(41,48);(2,6)(7,9)(8,13)(14,15)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,30)(27,32)(33,35)(34,39)(36,37)(38,45)(40,47)(41,46)(42,43)(44,48);(3,10)(4,12)(7,9)(8,13)(17,19,22,28)(18,24,25,23)(20,29,30,26)(21,27,32,31)(33,34,38,42)(35,39,45,43)(36,46,44,40)(37,41,48,47);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(33,36,38,44)(34,40,42,46)(35,37,45,48)(39,47,43,41);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,21)(19,26)(22,30)(23,27)(24,31)(25,32)(28,29)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,20)(19,27)(22,32)(23,26)(24,29)(25,30)(28,31)(33,37)(34,41)(35,44)(36,45)(38,48)(39,40)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669137);