Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2056 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)(13,14,18,22,17,21,16,26)(15,23,24,28,25,19,27,20);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,21,23)(16,27,18,24)(20,26,28,22);(6,9)(10,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,22)(21,28)(23,26);(5,7,8,11)(6,10,9,12)(14,21)(19,23)(20,28)(22,26);(13,16,17,18)(14,22,21,26)(15,24,25,27)(19,28,23,20)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,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342708);