Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394469 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2227 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)(7,14)(8,16)(9,17)(10,13)(11,15)(12,19)(18,20);(5,7)(6,11)(8,12)(9,18)(10,16)(13,20)(14,17)(15,19);(5,8,9,11,10,14,19,20)(6,7,12,15,13,16,17,18);(2,4)Polycyclic presentation:
2^9|c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72258208, #72258209, #72258213, … |
| maximal normal subgroup of | #72258208, #72258209, #72258213, … |
| maximal quotient of | #72258241, #72258255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301781);