Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(13,14,18,21)(15,20,24,19)(16,25,27,17)(22,23,26,28);(10,12)(13,15,18,24)(14,19,21,20)(16,22,27,26)(17,23,25,28);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,15)(14,20)(16,26)(17,23)(18,24)(19,21)(22,27)(25,28);(5,6,8,7)(13,16,18,27)(14,17,21,25)(15,22,24,26)(19,23,20,28);(1,3,4,2)(5,7,8,6)(13,17)(14,16)(15,23)(18,25)(19,22)(20,26)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280328, #72280735 |
| maximal normal subgroup of | #72280328, #72280735 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1316200);