Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,22)(19,20)(23,24);(5,7,8,6)(10,13)(14,16)(18,23)(19,20)(22,24);(2,4)(9,11,12,15)(10,14,13,16)(17,18,20,23,21,24,19,22);(10,13)(14,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72310231, #72310234, #72310243, … |
| maximal normal subgroup of | #72310231, #72310234, #72310243, … |
| maximal quotient of | #72310265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275403);