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 |
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 | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,14,18,15,19,13,24)(11,20,22,17,12,23,21,16);(2,4)(5,7,8,6)(9,11,14,22,15,12,13,21)(10,16,24,23,19,17,18,20);(1,2,3,4)(9,10)(11,16)(12,17)(13,18)(14,24)(15,19)(20,21)(22,23);(9,12,13,22,15,11,14,21)(10,17,18,23,19,16,24,20)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,c^2=g,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 | #72322397, #72322413, #72322424, … |
| maximal normal subgroup of | #72322397, #72322413, #72322424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277478);