Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C213×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C2 | C24×C42 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
14394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(1,2,3,4)(5,7,8,6)(10,14)(15,21)(16,19)(22,24);(11,17)(15,21)(18,23)(22,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,b^a=bg,f^a=fg,e^b=eg,d^c=dg260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532414);