Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,15,18,14,19,13,17)(11,12,16,22,20,23,21,24);(2,4)(9,11)(10,12)(13,16)(14,20)(15,21)(17,22)(18,24)(19,23);(6,8)(9,12)(10,16)(11,17)(13,24)(14,23)(15,22)(18,20)(19,21);(9,13,14,15)(10,17,19,18)(11,16,20,21)(12,22,23,24)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238738, #72238742, #72238755, … |
| maximal normal subgroup of | #72238738, #72238742, #72238755, … |
| maximal quotient of | #72238989 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264779);