Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2106 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)(13,14,18,21,19,22,17,20)(15,16,23,26,25,27,24,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,24)(18,23)(19,25)(20,28)(21,26)(22,27);(6,9)(10,12)(13,16)(14,15)(17,26)(18,28)(19,27)(20,23)(21,24)(22,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,20,22,21)(15,23,25,24)(16,26,27,28);(13,17,19,18)(14,21,22,20)(15,24,25,23)(16,26,27,28)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,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342690);