Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 |
| prime radical | #53202 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4068 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)(3,10)(4,13)(5,9)(6,8)(7,12)(11,15)(14,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,30)(29,31)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3)(2,7)(4,14)(5,11)(6,12)(8,10)(9,16)(13,15)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,29)(30,31)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(18,23)(22,25)(24,31)(30,32)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(2,8)(7,10)(9,13)(15,16)(18,23)(19,26)(24,31)(27,28);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,15,10,16)(17,20,21,29)(18,24,23,31)(19,27,26,28)(22,30,25,32)(34,40)(35,42)(39,48)(41,43);(1,5,6,4)(2,9,8,13)(3,11,12,14)(7,16,10,15)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013124);