Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,20)(17,22)(18,23)(21,25)(24,26)(27,28);(1,2,3,4)(13,14)(15,24)(16,27)(17,21)(18,23)(19,26)(20,28)(22,25);(5,7,8,12)(6,9,10,11)(13,15,25,28,18,16,17,26)(14,19,21,27,23,20,22,24);(13,16,25,26,18,15,17,28)(14,20,21,24,23,19,22,27);(13,17,18,25)(14,21,23,22)(15,26,16,28)(19,27,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,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, 6373438);