Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C26 | Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,11,14,18)(7,13,15,19,16,20,17,10);(5,7,14,15,9,16,8,17)(6,10,18,13,11,19,12,20);(6,11)(12,18);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72216238, #72216253, … |
| maximal normal subgroup of | #72215520, #72216238, #72216253, … |
| maximal quotient of | #72238751, C2×#396483, #72248342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303679);