Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,21)(16,24)(17,22)(18,19)(20,23)(25,28)(26,27);(1,2)(3,4)(6,9)(10,12)(13,15,17,18)(14,19,22,21)(16,25,26,23)(20,27,28,24);(2,4)(13,16)(14,20)(15,23)(17,26)(18,25)(19,24)(21,27)(22,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(16,26)(20,28)(23,25)(24,27);(2,4)(13,15,17,18)(14,21,22,19)(16,23,26,25)(20,27,28,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483247);