Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
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×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,19)(12,16)(13,22)(14,24)(15,18)(17,20)(21,23);(6,8)(9,11,20,18,14,21,23,24,15,17,19,10,12,22,13,16);(1,2,3,4);(9,12,15,14)(10,16,18,24)(11,21,17,22)(13,19,23,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224828, #72224833, #72224835, … |
| maximal normal subgroup of | #72224828, #72224833, #72224835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402319);