Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,22)(16,21)(17,19)(18,20)(23,28)(24,26)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,20,22)(16,24,25,23)(21,28,27,26);(14,20)(19,22)(21,27)(26,28);(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,28)(25,27);(13,16,18,25)(14,21,20,27)(15,23,17,24)(19,26,22,28);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,22,20,19)(16,24,25,23)(21,28,27,26)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532895);