Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1257 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)(5,6,8,10)(7,9,11,12)(13,14,20,15)(16,18,22,19)(17,21,26,23)(24,27,28,25);(5,7)(6,9)(8,11)(10,12)(13,15,18,22,20,14,19,16)(17,24,25,21,26,28,27,23);(7,11)(9,12)(13,16,19,14,20,22,18,15)(17,24,25,21,26,28,27,23);(2,4)(13,17)(14,21)(15,23)(16,24)(18,27)(19,25)(20,26)(22,28)Polycyclic presentation:
2^9|a^2=gi,b^2=e,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341101);