Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478211 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17136 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
899 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 899 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,21,27)(3,8,13,24,14,25,29,32)(4,15,5,18,17,26,20,9)(12,23,28,31,16,30,19,22);(1,3,6,13,7,14,21,29)(2,8,10,24,11,25,27,32)(4,16,20,28,17,12,5,19)(9,22,26,30,18,31,15,23);(1,4,6,20,7,17,21,5)(2,9,10,26,11,18,27,15)(3,12,13,19,14,16,29,28)(8,22,24,30,25,31,32,23);(1,5,6,17,7,20,21,4)(2,9,10,15,11,18,27,26)(3,12,13,28,14,16,29,19)(8,23,24,31,25,30,32,22)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ef,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251684, #72251708, … |
| maximal normal subgroup of | #72251676, #72251684, #72251708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385407);