Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| 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 | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,12,11)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27);(5,7,8,12)(6,10,9,11)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(1,2,3,4)(6,9)(7,11)(10,12);(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27);(6,9)(7,11)(10,12)(15,23)(18,21)(22,28)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591064);