Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10)(11,18)(12,16)(13,17,21,15)(14,19)(20,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,23,17,22);(2,5)(6,8)(10,15)(11,19)(13,21)(14,22)(16,17)(18,23);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,17)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403769);