Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3339 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,19)(17,25)(18,24)(20,22)(23,28)(26,27);(5,7)(6,9)(8,12)(10,11)(13,14,16,19)(15,18,23,26)(17,20,22,25)(21,27,28,24);(13,15,22,27,16,23,17,24)(14,18,25,28,19,26,20,21);(7,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28);(1,3,4,2)(6,10)(9,11)(14,19)(18,26)(20,25)(21,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310480);