Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500028 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,18,26,16,19,17,21)(15,22,25,20,23,28,24,27);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,19)(20,27)(21,26)(22,28);(5,7)(6,9)(8,12)(10,11)(13,15,16,23)(14,20,19,27)(17,25,18,24)(21,28,26,22);(7,12)(9,11);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6409818);