Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #52096 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18,13,16,22,24)(11,15,20,23,19,12,17,21);(1,2)(3,4)(9,11)(10,15)(12,16)(13,19)(14,20)(17,22)(18,23)(21,24);(6,8)(10,16)(11,19)(17,20)(18,24);(2,4)(10,16)(11,19)(17,20)(18,24);(9,12,14,21,13,15,22,23)(10,17,18,11,16,20,24,19)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=dfi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72343912, #72343972 |
| maximal normal subgroup of | #72343912, #72343972 |
| maximal quotient of | #72343927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290971);