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 | |
1420 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)(9,10,13,15)(11,16,12,14)(17,18,21,23)(19,26,30,32)(20,24,27,25)(22,28,31,29);(1,2,3,4)(5,6)(7,8)(17,19,27,28,21,30,20,29)(18,22,24,32,23,31,25,26);(6,8)(9,11,13,12)(10,14,15,16)(18,23)(22,31)(24,25)(26,32);(2,4)(9,12,13,11)(10,14,15,16)(18,24)(19,28)(20,27)(22,31)(23,25)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=fi,d^2=fi,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, 338246);