Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23426 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,20)(15,23,24,18)(17,21,27,22)(19,25,28,26);(6,9)(10,12)(13,14,16,20)(15,19,24,28)(17,22,27,21)(18,26,23,25);(2,4)(13,15)(14,18)(16,24)(17,26)(19,22)(20,23)(21,28)(25,27);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,24)(17,25)(18,22)(20,28)(21,23)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303220);