Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,18,19)(15,16,22,24)(17,20,25,26)(21,23,27,28);(2,4)(13,15)(14,16)(17,23)(18,22)(19,24)(20,27)(21,26)(25,28);(5,7,8,12)(6,9,10,11)(13,16,14,15,18,24,19,22)(17,23,20,21,25,28,26,27);(13,17,14,20,18,25,19,26)(15,21,24,28,22,27,16,23);(13,14,18,19)(15,16,22,24)(17,20,25,26)(21,23,27,28)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373209);