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 | |
3336 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)(13,14,17,18)(15,19,20,16)(21,22,25,24)(23,27,28,26);(2,4)(6,8)(15,20)(16,19)(21,23,25,28)(22,26,24,27);(9,10,12,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,6,7,8)(9,11,12,10)(13,16)(14,15)(17,19)(18,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=efg,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 | #72299406, #72299417, #72299488, … |
| maximal normal subgroup of | #72299406, #72299417, #72299488, … |
| maximal quotient of | #72299897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 65591);