Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| 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 | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,27)(19,26,21,30)(20,25,22,31)(24,29,32,28);(1,2)(3,4)(17,19,20,29,23,21,22,28)(18,24,31,30,27,32,25,26);(2,4)(10,13)(14,16)(17,20,23,22)(18,25,27,31)(19,28,21,29)(24,30,32,26);(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,26,27,30)(20,29,22,28)(24,31,32,25)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=i,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334731);