Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,12,11)(13,14,16,18)(15,17,19,20)(21,22,25,24)(23,27,28,26);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(23,28)(26,27);(1,2)(3,4)(13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,27)(25,28);(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,9)(8,12)(10,11)(13,15,16,19)(14,17,18,20)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 697721);