Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,17,14,18,13,19)(11,20,21,12,16,24,22,23);(5,6)(7,8)(9,11,14,16)(10,12,18,23)(13,21,15,22)(17,20,19,24);(2,4)(6,8)(9,12)(10,16)(11,18)(13,20)(14,23)(15,24)(17,22)(19,21);(2,4)(10,17)(11,21)(13,15)(16,22)(18,19)(20,24);(10,18)(12,23)(17,19)(20,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,e^c=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315616, #72315740, #72315905 |
| maximal normal subgroup of | #72315616, #72315740, #72315905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408022);