Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(10,13)(14,16)(17,18)(19,24)(20,27)(21,23)(22,30)(25,26)(28,31)(29,32);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(18,23)(24,30)(25,31)(27,32);(2,4)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,22)(18,24,23,30)(20,28,29,26)(25,27,31,32);(11,15)(14,16)(17,20,21,29)(18,25,23,31)(19,26,22,28)(24,32,30,27)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=i,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361901);