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 | |
4133 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)(3,9,11,4)(5,13,14,8)(10,12,16,15)(33,34,38,37)(35,40,43,39)(36,41,46,45)(42,48,44,47);(1,3,6,11)(2,4,7,9)(5,10,14,16)(8,12,13,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,38,43)(34,39,37,40)(36,44,46,42)(41,48,45,47);(2,7)(4,9)(5,14)(10,16)(18,22)(19,26)(20,29)(31,32)(33,35,38,43)(34,40,37,39)(36,44,46,42)(41,47,45,48);(1,2)(3,9)(4,11)(5,8)(6,7)(10,15)(12,16)(13,14)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,39)(36,44)(37,40)(38,43)(41,48)(42,46)(45,47);(1,4,6,9)(2,3,7,11)(5,12,14,15)(8,10,13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,38,46)(34,41,37,45)(35,42,43,44)(39,47,40,48);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,34)(35,40,43,39)(36,45,46,41)(42,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965391);