Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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×#3487 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10,13,17)(11,18,20,23)(12,16,14,21)(15,22,24,19);(1,2)(3,4)(9,10)(11,15)(12,21)(13,17)(14,16)(18,19)(20,24)(22,23);(9,11,13,20)(10,15,17,24)(12,19,14,22)(16,23,21,18);(2,4)(9,12,13,14)(10,16,17,21)(11,19,20,22)(15,23,24,18);(10,17)(15,24)(16,21)(18,23)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,e^b=ei,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72342372, #72342373, #72342376 |
| maximal normal subgroup of | #72342372, #72342373, #72342376 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286279);