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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1408 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)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,21,25,31)(20,27,29,26)(24,30,32,28);(1,2,3,4)(9,11)(10,13)(12,15)(14,16)(17,19,21,23,22,25,31,18)(20,24,28,27,29,32,30,26);(5,6,7,8)(11,15)(13,16)(18,23)(19,25)(24,32)(26,27);(2,4)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341068);