Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(10,14)(11,12)(15,16)(18,19)(20,24)(22,23);(1,2,3,4)(9,10,11,14,13,16,12,15)(17,18,20,19,21,23,24,22);(10,14)(11,12)(15,16)(17,19,20,18,21,22,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fi,d^c=dh,f^c=fh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217677 |
| maximal normal subgroup of | #72217677 |
| maximal quotient of | #72214372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28600);