Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(6,10)(11,12)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,17)(18,24)(20,22)(21,28)(23,25)(26,27);(5,8,9,7)(6,12,10,11)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28);(13,16,17,15)(14,19,20,22)(18,23,25,24)(21,28,27,26)Polycyclic presentation:
2^9|c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491789);