Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23);(5,7,8,6)(10,13)(14,16)(17,19,21,20)(18,22,23,24);(17,20,21,19)(18,22,23,24);(1,2,3,4);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514814);