Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | C2×#3045 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,14,19)(11,15,13,18)(12,17,22,23)(16,21,24,20);(1,3,4,2)(6,7)(9,11,14,13)(10,15,19,18)(12,21,22,20)(16,17,24,23);(9,12)(10,16)(11,20)(13,21)(14,22)(15,23)(17,18)(19,24);(5,7,8,6)(9,12)(10,17)(11,21)(13,20)(14,22)(15,24)(16,18)(19,23);(9,11,14,13)(10,18,19,15)(12,20,22,21)(16,17,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284272, #72284863, #72285294, … |
| maximal normal subgroup of | #72284272, #72284863, #72285294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289091);