Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,17,14,12,22,16)(11,15,21,18,19,23,20,24);(1,2,3,4)(9,11)(10,15)(12,23)(13,21)(14,19)(16,24)(17,18)(20,22);(6,8)(9,12)(10,14)(11,18)(13,16)(15,20)(17,22)(19,24)(21,23);(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238302, #72239934, #72262937, … |
| maximal normal subgroup of | #72238302, #72239934, #72262937, … |
| maximal quotient of | #72238285, #72239947, #72321607 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264934);