Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(9,10,12,11)(14,17)(18,20)(23,27)(25,28);(2,4)(6,7)(10,11)(14,17)(15,19)(21,22,24,26)(23,28,27,25);(5,6,8,7)(9,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,8,6)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,25,26,28)Polycyclic presentation:
2^9|a^2=e,b^2=efg,c^2=fgh,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 65498);