Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,7,8)(14,17)(15,19)(22,25)(23,27);(10,12)(15,19)(18,20)(21,22,24,25)(23,26,27,28);(9,10)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,8)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72278676, #72278732, C2×#156478, … |
| maximal normal subgroup of | #72278676, #72278732, C2×#156478, … |
| maximal quotient of | C2×#156478, #72299908 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 63674);