Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(10,11)(13,14,18,19)(15,20,17,16)(21,22,25,26)(23,27,28,24);(5,6,9,7)(8,11,12,10)(13,15,18,17)(14,16,19,20)(22,26)(24,27);(5,6,9,7)(8,10,12,11)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,17,18,15)(14,16,19,20)(21,23)(22,24)(25,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fh,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72263363, #72263366, #72263389, … |
| maximal normal subgroup of | #72263363, #72263366, #72263389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 559781);