Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,13,18,14,19,24,17)(11,20,12,15,22,16,23,21);(1,2,3,4)(5,6)(7,8)(9,11,13,12,14,22,24,23)(10,15,17,20,19,21,18,16);(2,4)(6,8)(9,10)(11,21)(12,16)(13,17)(14,19)(15,22)(18,24)(20,23);(9,12,13,22,14,23,24,11)(10,16,17,15,19,20,18,21)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320970, #72320995, #72321053, … |
| maximal normal subgroup of | #72320970, #72320995, #72321053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274273);