Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1438 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)(13,14,19,21,18,23,17,22)(15,24,25,20,27,28,26,16);(1,2)(3,4)(9,10)(11,12)(13,15)(14,20)(16,23)(17,25)(18,27)(19,26)(21,24)(22,28);(2,4)(9,10,12,11)(13,16,18,20)(14,15,23,27)(17,28,19,24)(21,25,22,26);(6,8)(9,11,12,10)(14,21)(15,25)(17,19)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=efi,c^2=fi,d^2=f,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, 338133);