Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
5022 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)(13,14,16,17)(15,18,20,19)(23,27)(25,28);(2,4)(6,8)(14,17)(18,19)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,25)(24,26)(27,28);(13,15)(14,18)(16,20)(17,19)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313592, #72313643, #72328380, … |
| maximal normal subgroup of | #72313592, #72313643, #72328380, … |
| maximal quotient of | #72314205 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 62875);