Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,27)(26,28);(5,7,9,12)(6,10,8,11)(16,25)(18,26)(19,24)(23,28);(13,15)(14,18,20,26)(16,23)(17,22)(19,21,24,27)(25,28);(2,4)(5,8,9,6)(7,10,12,11)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 350717);