Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| 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 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(17,18)(19,23)(20,22)(21,24)(25,28)(26,29)(27,30)(31,32);(1,2,3,4)(9,10)(11,14)(12,13)(15,16)(18,22)(19,26)(24,25)(27,32);(6,8)(10,13)(14,16)(17,19,20,26)(18,23,22,29)(21,27,28,32)(24,30,25,31);(9,11,12,15)(10,14,13,16)(18,24)(21,28)(22,25)(23,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=i,d^2=gi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311515);