Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,19)(15,23,25,28)(16,20,18,22)(21,26,27,24);(6,9)(10,12)(14,19)(20,22)(21,27)(23,28);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(16,18)(19,22)(21,28)(23,27)(24,26);(13,15)(14,21)(16,24)(17,25)(18,26)(19,27)(20,28)(22,23);(13,16,17,18)(14,22,19,20)(15,24,25,26)(21,23,27,28)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,h^a=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536448);