Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432833 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,17,21,18,20,16,25)(15,22,24,27,23,28,26,19);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(15,23)(19,27)(22,28)(24,26);(6,9)(10,12)(13,15)(14,19)(16,24)(17,26)(18,23)(20,27)(21,28)(22,25);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,25)(22,28);(13,16,18,17)(14,21,20,25)(15,24,23,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342685);