Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,12,7,11,13,10,15)(8,18,14,20,17,9,19,16);(5,7)(6,10)(8,9)(11,15)(12,13)(14,16)(17,18)(19,20);(2,4)(6,13)(7,15)(9,18)(16,20);(5,8)(6,9)(7,16)(10,19)(11,17)(12,14)(13,18)(15,20);(5,9,12,16,11,18,10,20)(6,14,7,17,13,19,15,8)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72315328, #72315597, #72315633, … |
| maximal normal subgroup of | #72315328, #72315597, #72315633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291060);