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 | C23×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
13206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(5,6,8,7)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22);(1,2,3,4)(5,7,8,6)(10,12)(15,18)(16,20)(22,24);(11,17)(15,18)(19,23)(22,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=g,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, 7532416);