Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3277 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2011 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)(7,10)(8,9)(11,12)(14,16)(20,24)(21,25)(22,26);(1,2,4,3)(6,9)(10,12)(13,14,18,16)(15,22,17,26)(19,25,28,21)(20,27,24,23);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15,18,17)(14,20,16,24)(19,27,28,23)(21,26,25,22);(13,16,18,14)(15,22,17,26)(19,21,28,25)(20,27,24,23);(13,17,18,15)(14,20,16,24)(19,23,28,27)(21,26,25,22)Polycyclic presentation:
2^9|b^2=gi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486473);