Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 | C2×#3150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,20,19,22,17,21)(15,23,24,27,25,26,28,16);(7,8)(10,12)(13,15)(14,16)(17,24)(18,28)(19,25)(20,26)(21,23)(22,27);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,15)(17,26)(18,23)(19,27)(20,24)(21,28)(22,25);(5,8,9,7)(6,10,11,12)(13,17,19,18)(14,20,22,21)(15,24,25,28)(16,26,27,23);(13,17,19,18)(14,21,22,20)(15,24,25,28)(16,23,27,26)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370204);