Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
5658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,4,8)(10,14)(11,12)(17,18,22,21)(19,24,20,23);(2,6)(7,8)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,22)(21,24);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,19)(18,23,21,24);(9,12,13,11)(10,15,14,16);(1,3,5,4)(2,7,6,8)(17,21,22,18)(19,24,20,23);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7545714);