Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394335 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1590 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)(6,8)(9,10,12,17,13,16,20,14)(11,15,19,22,18,23,24,21);(1,2,4,3)(10,14)(11,18)(12,20)(15,22)(16,17)(21,23);(1,3,4,2)(9,11,12,19,13,18,20,24)(10,15,17,22,16,23,14,21);(5,6,7,8)(10,16)(14,17)(15,23)(21,22)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=hi,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, 320300);