Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | |
2366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,24)(17,23)(18,22)(20,26)(21,25)(27,28);(1,2,4,3)(6,9)(10,12)(13,15)(14,19)(16,27)(17,26)(18,25)(20,23)(21,22)(24,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,27)(17,26)(18,25)(19,23)(21,24)(22,28);(13,15)(14,21)(16,25)(17,26)(18,27)(19,22)(20,24)(23,28);(13,16,17,18)(14,22,23,24)(15,25,26,27)(19,28,20,21)Polycyclic presentation:
2^9|b^2=gi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,e^b=ei,e^c=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536657);