Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
883 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)(7,8,17,13)(9,12)(10,16)(11,15)(14,19,20,18);(1,2)(3,4)(6,12,16,15)(7,17)(8,18,13,19)(14,20);(2,4)(5,7,9,14,10,17,11,20)(6,13,12,18,16,8,15,19);(5,8)(6,14)(7,15)(9,19)(10,13)(11,18)(12,17)(16,20)Polycyclic presentation:
2^9|a^2=efh,b^2=f,c^2=ef,e^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^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321897 |
| maximal normal subgroup of | #72321897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357830);