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 | |
4153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(18,19)(20,27)(24,26)(28,32)(33,34)(35,36)(37,46)(38,41)(39,42)(40,43)(44,47)(45,48);(1,2,6,7)(3,5,9,14)(4,8,13,11)(10,12,15,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,37)(36,43)(38,44)(39,45)(40,46)(41,42)(47,48);(2,7)(5,14)(8,11)(12,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,36,40,37)(34,35,43,46)(38,47,45,42)(39,41,44,48);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,37,40,36)(34,35,43,46)(38,42,45,47)(39,41,44,48);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,38,40,45)(34,41,43,48)(35,44,46,39)(36,47,37,42);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,39,40,44)(34,42,43,47)(35,45,46,38)(36,48,37,41)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8094028);