Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,13)(12,20)(14,18)(15,17)(16,22)(19,23)(21,24);(2,4)(9,11,13,17,15,18,14,10)(12,19,22,24,23,20,21,16);(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20);(10,17)(11,18)(16,24)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238300, #72238841, #72239933, … |
| maximal normal subgroup of | #72238300, #72238841, #72239933, … |
| maximal quotient of | #72238285, #72238846, #72321690 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260575);