Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,9)(7,11,8,12)(14,18)(15,20)(21,22)(23,25)(24,26)(27,28);(6,9)(11,12)(21,22,24,26)(23,27,28,25);(5,7,10,8)(6,11,9,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15,17,20)(14,19,18,16);(5,9,10,6)(7,12,8,11)(13,16,17,19)(14,20,18,15)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 559937);