Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2016 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)(9,10)(11,15)(12,13)(14,16)(17,18,23,26)(19,27,29,28)(20,21,25,22)(24,31,32,30);(1,2,3,4)(5,6)(7,8)(10,13)(14,15)(17,19,21,30,23,29,22,31)(18,24,20,27,26,32,25,28);(6,8)(17,20,22,18,23,25,21,26)(19,28,31,32,29,27,30,24);(9,11,12,16)(10,14,13,15)(19,29)(24,32)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=f,d^2=h,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=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323795);