Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394335 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3055 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,8)(9,10,13,19,15,11,14,17)(12,18,20,23,22,16,21,24);(1,2,4,3)(9,11,15,10)(12,18,22,16)(13,19,14,17)(20,24,21,23);(1,3,4,2)(9,12,14,21,15,22,13,20)(10,16,17,23,11,18,19,24);(6,8)(9,13,15,14)(10,17,11,19)(12,20,22,21)(16,23,18,24)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304411);