Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100203 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator, prime radical | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,3)(5,6,7,10,8,9,11,12)(13,14)(15,18)(16,23,24,19)(17,20)(21,25,28,27)(22,26);(1,2,4,3)(13,15)(14,18)(16,21)(17,22)(19,27)(20,26)(23,25)(24,28);(1,3,4,2)(6,9)(7,11)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27214);