Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,8)(10,11)(13,14,18,21,17,20,16,22)(15,23,24,19,26,28,25,27);(1,2)(3,4)(9,10)(11,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(2,4)(9,10,12,11)(14,20)(19,27)(21,22)(23,28);(6,8)(9,11,12,10)(14,21)(15,24)(16,18)(20,22)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=ef,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338130);