Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(14,20)(15,23)(22,28)(25,26);(1,2,3,4)(13,14,18,20)(15,24,23,27)(16,26,21,25)(17,22,19,28);(6,9)(10,12)(13,15)(14,21)(16,20)(17,26)(18,23)(19,25)(22,27)(24,28);(5,7,8,11)(6,10,9,12)(13,16,18,21)(14,15,20,23)(17,27,19,24)(22,25,28,26);(13,17,18,19)(14,22,20,28)(15,25,23,26)(16,27,21,24)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357205);