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 | |
4081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,39,40)(35,43,37,41)(36,45,46,42)(38,44,48,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(34,40)(36,46)(41,43)(44,47);(4,12)(7,14)(10,16)(13,15)(19,25)(22,31)(27,29)(28,32)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,42,48,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36)(34,42)(35,44)(37,47)(38,43)(39,46)(40,45)(41,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,42,48,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,38,39,48)(34,44,40,47)(35,45,37,42)(36,43,46,41)Polycyclic presentation:
2^9|a^2=gi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230505);