Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,19)(18,22)(20,23)(21,24)(25,27)(26,28);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,24,20,26,19,25)(14,21,16,17,23,27,22,28);(7,11)(9,12)(13,16,18,23,20,22,19,14)(15,17,24,27,26,28,25,21);(7,11)(9,12)(13,17)(14,15)(16,24)(18,27)(19,21)(20,28)(22,25)(23,26);(13,18,20,19)(14,22,23,16)(15,24,26,25)(17,21,28,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6364570);