Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395781 |
| 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)(5,6,8,13,10,14,9,11)(7,15,16,20,18,19,17,12);(6,11)(7,16)(8,9)(13,14)(15,19)(17,18);(5,7,8,17,10,18,9,16)(6,12,13,19,14,20,11,15);(2,4)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72216142, #72216217, … |
| maximal normal subgroup of | #72215520, #72216142, #72216217, … |
| maximal quotient of | #72238757, C2×#395781, #72248329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302977);