Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60790 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#432 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10,15,18,14,19,13,24)(11,20,17,23,21,16,22,12);(1,3,6,2)(4,7,5,8)(9,11,13,22,14,21,15,17)(10,16,24,23,19,20,18,12);(9,12,15,20,14,23,13,16)(10,17,18,21,19,22,24,11);(1,4,6,5)(2,7,3,8)(9,13,14,15)(10,18,19,24)(11,17,21,22)(12,16,23,20);(1,5,6,4)(2,7,3,8)(9,13,14,15)(10,18,19,24)(11,17,21,22)(12,16,23,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6302959);