Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1959 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)(7,10)(8,9)(11,12)(13,14,19,17,20,15,18,22)(16,21,28,25,26,23,27,24);(1,2)(3,4)(13,15,20,14)(16,24,26,25)(17,18,22,19)(21,28,23,27);(2,4)(6,9)(10,12)(13,16,18,27,20,26,19,28)(14,21,22,24,15,23,17,25);(2,4)(13,17,19,15,20,22,18,14)(16,25,28,23,26,24,27,21);(5,7,8,11)(6,10,9,12)(16,26)(21,23)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=f,d^2=fi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363847);