Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,8)(9,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,25)(23,26,28,27);(2,4)(6,8)(9,10,11,12)(14,17)(15,19)(22,25)(26,27);(9,10)(11,12)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,17)(18,19)(21,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72329882, #72329925, #72330036, … |
| maximal normal subgroup of | #72329882, #72329925, #72330036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 66066);