Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #44973 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2037 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,28)(22,25)(23,26)(24,27);(1,2)(3,4)(6,9)(10,11)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(2,4)(13,14)(15,19)(16,18)(17,20)(21,28)(22,25)(23,26)(24,27);(2,4)(13,15,16,22)(14,19,18,25)(17,24,23,21)(20,27,26,28);(5,7,8,12)(6,10,9,11)(14,18)(19,25)(20,26)(27,28)Polycyclic presentation:
2^9|d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425675);