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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,16)(11,15,12,14)(17,18,22,23)(19,21,20,24);(6,8)(18,23)(21,24);(5,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,22,20)(18,21,23,24);(1,2)(3,4)(9,12,13,11)(10,15,16,14)(17,20,22,19)(18,21,23,24);(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=fg,c^2=fgh,d^2=fg,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 | #72331635, #72331685, #72331701, … |
| maximal normal subgroup of | #72331635, #72331685, #72331701, … |
| maximal quotient of | #72332755, #72333280 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1327678);