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 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5608 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)(13,14,19,22)(15,20,17,16)(18,26,27,21)(23,24,25,28);(10,12)(13,15,19,17)(14,16,22,20)(18,23,27,25)(21,24,26,28);(5,6,8,7)(9,10)(11,12)(13,16)(14,15)(17,22)(18,24)(19,20)(21,23)(25,26)(27,28);(1,2,3,4)(5,7,8,6)(13,17,19,15)(14,20,22,16)(18,25,27,23)(21,28,26,24);(13,18,19,27)(14,21,22,26)(15,23,17,25)(16,24,20,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,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 | #72280804 |
| maximal normal subgroup of | #72280804 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 548199);