Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401626 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#397 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2344 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)(5,6,9,13,10,14,8,12)(7,15,17,19,18,20,16,11);(5,7,8,16,10,18,9,17)(6,11,13,15,14,19,12,20);(6,12)(7,16)(8,9)(11,19)(13,14)(17,18);(2,4)Polycyclic presentation:
2^9|a^2=g,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,f^b=fi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72216029, #72216037, … |
| maximal normal subgroup of | #72215520, #72216029, #72216037, … |
| maximal quotient of | #72238753, #72238757, C2×#401626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308822);