Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1019 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)(9,10,15,19,14,20,13,18)(11,21,23,24,22,17,12,16);(1,2)(3,4)(9,11,13,12,14,22,15,23)(10,16,19,21,20,24,18,17);(2,4)(6,8)(9,12,13,22,14,23,15,11)(10,17,19,16,20,21,18,24);(10,18,20,19)(11,22)(12,23)(16,21,24,17)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309702, #72310129, #72310141, … |
| maximal normal subgroup of | #72309702, #72310129, #72310141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284869);