Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(9,10)(11,12)(13,14,18,22)(15,23,26,27)(16,17,19,21)(20,24,28,25);(5,6)(7,8)(13,15,18,26)(14,20,22,28)(16,24,19,25)(17,23,21,27);(2,4)(6,8)(13,16,18,19)(14,21,22,17)(15,24,26,25)(20,27,28,23);(10,12)(13,17,19,22,18,21,16,14)(15,20,24,27,26,28,25,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327064);