Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,7)(9,10)(11,12)(13,14,16,15)(17,18,20,22)(19,21,23,24);(5,6,8,7)(9,10,12,11)(13,15,16,14)(19,23)(21,24);(6,7)(10,11)(17,19,20,23)(18,21,22,24);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de104 conjugacy classes — view the full character table.
| maximal subgroup of | #72298126, #72298237, #72298352, … |
| maximal normal subgroup of | #72298126, #72298237, #72298352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 63220);