Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | #25723 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,21,17,19,24,18)(15,20,23,27,22,26,28,25);(2,3)(14,18)(16,24)(19,21)(20,25)(23,28)(26,27);(1,2,4,3)(6,9)(10,12)(14,19)(18,21)(20,26)(25,27);(1,3,4,2)(13,15,17,22)(14,20,19,26)(16,23,24,28)(18,25,21,27);(5,7,8,11)(6,10,9,12)(15,22)(20,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374779);