Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #299067 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(10,11)(14,16)(17,18,20,24,21,19,22,23);(1,3,5,4)(2,6,8,7)(9,10,13,14,12,11,15,16)(17,19,20,23,21,18,22,24);(9,11)(10,12)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23);(1,4,5,3)(2,7,8,6)(17,20,21,22)(18,23,19,24)Polycyclic presentation:
2^9|a^2=ef,b^2=eghi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226827 |
| maximal normal subgroup of | #72226827 |
| maximal quotient of | #72226806 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 230971);