Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,7,9,8,10,11,12)(13,14,17,21,15,18,24,22)(16,25,27,20,23,19,26,28);(1,2,3,4)(14,18)(19,25)(20,28)(21,22);(14,19)(16,26)(17,24)(18,25)(20,21)(22,28)(23,27)Polycyclic presentation:
2^9|a^2=f,b^2=di,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^c=egi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318674, #72318679, #72318709, … |
| maximal normal subgroup of | #72318674, #72318679, #72318709, … |
| maximal quotient of | #72322636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41560);