Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,11,15,10,13,8,18)(7,12,16,20,17,14,9,19);(1,2)(3,4)(5,7,11,9,10,17,8,16)(6,12,15,19,13,14,18,20);(2,4)(5,8,10,11)(6,13)(7,16,17,9)(12,14)(15,18)(19,20);(5,9,11,17,10,16,8,7)(6,14,15,20,13,12,18,19)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321148 |
| maximal normal subgroup of | #72321148 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358030);