Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,17,22,18,20,27,28)(15,23,24,16,21,25,26,19);(6,10)(8,11)(13,15,18,21)(14,19,20,16)(17,26,27,24)(22,25,28,23);(2,4)(5,6)(7,8)(9,10)(11,12)(14,20)(15,21)(22,28)(24,26);(5,7,9,12)(6,11,10,8)(13,16,18,19)(14,21,20,15)(17,25,27,23)(22,26,28,24);(5,8,9,11)(6,7,10,12)(15,21)(16,19)(23,25)(24,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524733);