Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,12,11)(13,14,18,21,17,20,16,22)(15,23,26,19,25,28,24,27);(1,2,4,3)(5,6)(7,8)(13,15,17,25)(14,19,20,27)(16,26,18,24)(21,23,22,28);(1,2,4,3)(6,8)(14,20)(19,27)(21,22)(23,28);(1,3,4,2)(9,11,12,10)(14,21)(16,18)(19,23)(20,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fi,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284566);