Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14,18,23)(15,24,19,20)(16,26,17,28)(21,25,22,27);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,18,19)(14,20,23,24)(16,25,17,27)(21,28,22,26);(2,4)(5,8,9,6)(7,11,12,10)(13,16,18,17)(14,21,23,22)(15,25,19,27)(20,28,24,26);(7,12)(10,11)(13,17,18,16)(14,22,23,21)(15,25,19,27)(20,28,24,26)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325434);