Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,12,15)(11,17,14,22)(13,16,20,23)(18,19,24,21);(1,2,3,4)(10,15)(16,23)(17,22)(18,24);(10,16)(11,14)(13,21)(15,23)(17,18)(19,20)(22,24);(5,7,8,6)(9,11,12,14)(10,17,15,22)(13,19,20,21)(16,24,23,18)Polycyclic presentation:
2^9|a^2=g,b^2=ei,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318530, #72318536, #72318577, … |
| maximal normal subgroup of | #72318530, #72318536, #72318577, … |
| maximal quotient of | #72322658 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403012);