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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2066 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)(7,11)(8,9)(10,12)(13,14)(15,16)(17,21)(18,22)(19,28)(20,23)(24,26)(25,27);(2,3)(6,9)(10,11)(13,15,19,21,20,25,18,24)(14,17,22,16,23,26,28,27);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,18)(19,26)(20,27)(21,22)(23,25)(24,28);(1,3,4,2)(13,17,20,26)(14,21,23,24)(15,22,25,28)(16,18,27,19);(13,18,20,19)(14,22,23,28)(15,24,25,21)(16,26,27,17)Polycyclic presentation:
2^9|b^2=fg,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373753);