Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481373 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,12,16,22,21,24,19,14,17,23,15,13,11,18,20);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(9,11)(10,13)(12,16)(14,21)(15,23)(17,22)(18,19)(20,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265009, #72265019, #72265070, … |
| maximal normal subgroup of | #72265009, #72265019, #72265070, … |
| maximal quotient of | #72309966, #72309990, C2×#496497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403693);