Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393963 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,24)(22,23);(9,11)(10,15)(12,19)(13,22)(14,20)(16,24)(17,23)(18,21);(2,4)(12,13)(15,23)(16,21)(19,22);(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,24,23,18)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279018, #72283333, #72287221, … |
| maximal normal subgroup of | #72279018, #72283333, #72287221, … |
| maximal quotient of | #72284659, #72284678 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301161);