Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #635984 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6084 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)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28);(10,12)(14,18)(19,21)(20,24)(26,27);(5,6,8,7)(9,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(1,2,3,4)(5,7,8,6)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27);(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^9|c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280694, #72280784, #72280858, … |
| maximal normal subgroup of | #72280694, #72280784, #72280858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 548197);