Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 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)(7,10)(8,9)(11,12)(13,14,15,18,17,20,23,21)(16,19,25,24,26,28,22,27);(1,2)(3,4)(6,9)(7,11)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,28)(26,27);(2,4)(6,9)(10,12)(13,15,17,23)(14,18,20,21)(16,22,26,25)(19,27,28,24);(5,7,8,11)(6,10,9,12)(13,16,17,26)(14,19,20,28)(15,22,23,25)(18,27,21,24)Polycyclic presentation:
2^9|a^2=ef,c^2=i,d^2=fi,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, 335587);