Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2010 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)(13,14,17,21)(15,22,24,27)(16,20,18,25)(19,26,28,23);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,25)(17,21)(18,20)(22,23)(24,28)(26,27);(6,9)(7,11)(13,15,16,26,17,24,18,23)(14,19,25,27,21,28,20,22);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,21,25)(15,23,24,26)(19,27,28,22);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426793);