Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,11,12)(13,14)(15,22)(16,21)(17,26)(18,27)(19,23)(20,24)(25,28);(5,6)(7,8)(13,15,19,25)(14,20,23,26)(16,24,18,17)(21,28,27,22);(1,2,4,3)(6,8)(13,16,19,18)(14,21,23,27)(15,24,25,17)(20,28,26,22);(1,3,4,2)(13,17,18,15,19,24,16,25)(14,22,21,26,23,28,27,20)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292602);