Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15172 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(13,14,19,21)(15,22,16,18)(17,25,27,20)(23,26,24,28);(10,12)(13,15,19,16)(14,18,21,22)(17,23,27,24)(20,26,25,28);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,16,19,15)(14,18,21,22)(17,23,27,24)(20,28,25,26);(5,6,8,7)(13,17,19,27)(14,20,21,25)(15,23,16,24)(18,26,22,28);(1,3,4,2)(5,7,8,6)(13,18)(14,15)(16,21)(17,26)(19,22)(20,23)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1099778);