Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4161 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)(3,11)(4,9)(5,10)(6,8)(7,13)(12,16)(14,15)(17,18)(19,25)(20,22)(21,28)(23,26)(24,30)(27,31)(29,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,16)(10,15)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,8)(7,11)(9,10)(15,16)(18,20)(19,26)(24,28)(27,29)(35,42)(36,45)(38,47)(39,43);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32)(34,40)(35,42)(39,43)(41,44);(1,5,6,4)(2,10,8,9)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|c^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012749);