Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,19,25,28)(16,20,18,21)(23,26,27,24);(6,9)(10,12)(13,14)(15,23)(16,21)(17,22)(18,20)(19,24)(25,27)(26,28);(2,4)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,22,21)(15,24,25,26)(19,23,28,27);(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,27,28,23)Polycyclic presentation:
2^9|a^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339480);