Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,12)(10,11)(13,14)(15,18)(16,20)(17,19)(22,25)(23,27);(5,7)(6,8)(9,11)(10,12)(13,15,18,14)(16,20,17,19)(21,22)(23,28)(24,25)(26,27);(13,16,18,17)(14,19,15,20)(22,25)(26,28);(5,8,10,9)(6,7,11,12)(13,17,18,16)(14,19,15,20)(21,23,24,27)(22,26,25,28);(5,9,10,8)(6,7,11,12)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428158);