Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,7,8)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,31)(30,32);(1,2,4,3)(9,10,12,14)(11,15,16,13)(17,19,22,28)(18,23,26,32)(20,30,21,29)(24,31,25,27);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(20,21)(23,27)(25,26)(31,32);(1,3,4,2)(18,25)(19,28)(20,21)(23,31)(24,26)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=gi,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270336);