Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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 | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3008 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)(7,9)(8,10)(11,12)(13,14)(15,22)(16,27)(17,21)(18,20)(19,26)(23,25)(24,28);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,25)(17,26)(18,28)(20,23)(21,22)(24,27);(6,10)(9,12)(13,14)(15,23)(16,20)(17,21)(18,27)(19,28)(22,25)(24,26);(13,14)(15,24)(16,20)(17,21)(18,27)(19,25)(22,28)(23,26);(13,16,17,18)(14,20,21,27)(15,25,26,28)(19,23,22,24)Polycyclic presentation:
2^9|a^2=fi,b^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347700);