Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,18,22)(15,24,16,20)(17,25,23,19)(21,26,28,27);(2,4)(5,7,9,8)(6,10,12,11)(13,15,18,16)(14,20,22,24)(17,28,23,21)(19,27,25,26);(13,16,19,27,18,15,25,26)(14,21,23,20,22,28,17,24);(1,2,3,4)(13,17,19,14,18,23,25,22)(15,20,26,28,16,24,27,21);(5,8,9,7)(6,11,12,10)(14,22)(17,23)(20,24)(21,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=h,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521376);