Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2424 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,21)(20,22)(23,24);(10,15)(13,16)(17,19)(18,20)(21,24)(22,23);(1,2,4,3)(9,11,14,12)(10,16,15,13)(17,20,23,21)(18,19,24,22);(5,6)(7,8)(9,12,14,11)(10,16,15,13)(17,21,23,20)(18,19,24,22);(1,3,4,2)(9,13,14,16)(10,12,15,11)(17,22,23,19)(18,21,24,20)Polycyclic presentation:
2^9|a^2=fhi,c^2=fgh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72333583, #72333625, #72333705 |
| maximal normal subgroup of | #72333583, #72333625, #72333705 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1341369);