Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,12,10,14,8,13)(7,15,18,20,19,16,17,11);(5,7,8,17,10,19,9,18)(6,11,12,15,14,20,13,16);(5,6)(7,16)(8,12)(9,13)(10,14)(11,17)(15,19)(18,20);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238735 |
| maximal normal subgroup of | #72238735 |
| maximal quotient of | #72238756 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302748);