Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394183 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,29,30,25)(20,32,31,24)(21,27,22,26);(1,2)(3,4)(10,13)(14,16)(17,19,21,31,23,30,22,20)(18,24,26,29,28,32,27,25);(2,4)(5,6,8,7)(17,20,22,30,23,31,21,19)(18,25,27,32,28,29,26,24);(5,7,8,6)(9,11,12,15)(10,14,13,16)(19,30)(20,31)(24,32)(25,29)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=f,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316273);