Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,11)(7,10,8,12)(13,14,16,22,19,15,18,20)(17,21,27,25,26,23,24,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,14)(16,22,18,20)(17,25,26,28)(21,24,23,27);(2,4)(6,11)(10,12)(13,16,19,18)(14,20,15,22)(17,24,26,27)(21,25,23,28);(5,8,9,7)(6,12,11,10)(13,17,19,26)(14,21,15,23)(16,24,18,27)(20,25,22,28)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337122);