Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 | |
2134 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)(13,14,18,22)(15,24,26,19)(16,20,25,28)(17,23,27,21);(2,4)(13,14)(15,20)(16,24)(17,21)(18,22)(19,25)(23,27)(26,28);(5,7,8,12)(6,9,10,11)(13,15,17,25,18,26,27,16)(14,19,23,20,22,24,21,28);(13,16,27,26,18,25,17,15)(14,20,21,19,22,28,23,24);(13,17,18,27)(14,21,22,23)(15,25,26,16)(19,28,24,20)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373200);