Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184831 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#95 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(13,14)(15,18)(16,23,24,19)(17,20)(21,27,28,25)(22,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,27)(24,28);(1,2,4,3)(6,9)(10,12)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(15,22)(18,26)(21,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270720);