Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,7)(8,11,12,10)(13,14,16,19)(15,21,22,18)(17,20,25,28)(23,27,24,26);(1,2,4,3)(5,7,9,6)(8,10,12,11)(13,15,16,22)(14,18,19,21)(17,23,25,24)(20,26,28,27);(5,8,9,12)(6,10,7,11)(15,22)(18,21)(23,24)(26,27);(6,7)(10,11)(15,23)(17,25)(18,26)(20,28)(21,27)(22,24);(1,3,4,2)(14,19)(15,22)(20,28)(23,24)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7452946);