Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,18,20,14)(12,15)(13,17)(16,21)(19,22,24,23);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,20)(15,23)(16,22)(17,18)(21,24);(3,7)(5,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,17)(18,23)(20,24);(9,12)(10,16,17,15)(11,19)(13,21)(14,22,18,23)(20,24)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225695, #72225701, #72225704, … |
| maximal normal subgroup of | #72225695, #72225701, #72225704, … |
| maximal quotient of | #72268221 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374325);