Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1156 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)(15,17)(16,20)(18,23)(19,21)(22,25)(24,26)(27,28);(5,6)(7,8)(13,15,17,14,19,23,18,21)(16,24,25,28,27,22,26,20);(1,2,4,3)(6,8)(13,16,19,27)(14,20,21,28)(15,22,23,24)(17,25,18,26);(1,3,4,2)(13,17)(15,23)(16,25)(18,19)(22,24)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292424);