Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3982 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)(9,10)(11,17,18,24)(12,16)(13,15)(14,19,22,21)(20,23);(2,4)(5,6,7,8)(12,20)(16,23)(17,24)(19,21);(6,8)(9,11)(10,14)(12,21)(13,18)(15,22)(16,24)(17,23)(19,20);(2,4)(10,15)(12,20)(14,22)(19,21)Polycyclic presentation:
2^9|a^2=h,b^2=g,f^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257936, #72257937, #72257941, … |
| maximal normal subgroup of | #72257936, #72257937, #72257941, … |
| maximal quotient of | #72265432, #72265739 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268866);