Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10,13,19)(11,17,21,15)(12,18,14,23)(16,22,24,20);(1,2,3,4)(9,11)(10,15)(12,22)(13,21)(14,20)(16,18)(17,19)(23,24);(9,11)(10,16)(12,20)(13,21)(14,22)(15,18)(17,23)(19,24);(5,7,8,6)(9,11)(10,17)(12,22)(13,21)(14,20)(15,19)(16,23)(18,24);(9,12,13,14)(10,18,19,23)(11,20,21,22)(15,24,17,16)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280029);