Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| 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 | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3732 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)(10,11)(13,14)(15,19)(16,20)(17,18)(22,23)(26,27);(2,4)(6,8)(10,11)(14,18)(16,19)(21,22,25,23)(24,27,28,26);(9,10,12,11)(13,15,17,20)(14,19,18,16)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,6,7,8)(9,11,12,10)(13,16,17,19)(14,20,18,15)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|b^2=fg,c^2=eg,d^2=eh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72328405, #72328505, #72328701, … |
| maximal normal subgroup of | #72328405, #72328505, #72328701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 65608);