Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356478 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13809 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#338 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1129 conjugacy classes of subgroups; only the 380 normal subgroups are shown.
This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 380 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:
(1,2,7,13,8,14,6,12)(3,15,18,27,20,26,19,9)(4,21,22,25,23,5,11,10)(16,17,28,24,29,31,32,30);(1,3,6,18,8,20,7,19)(2,9,13,26,14,27,12,15)(4,16,11,28,23,29,22,32)(5,24,10,17,21,30,25,31);(1,4,6,11,8,23,7,22)(2,10,12,5,14,25,13,21)(3,16,18,28,20,29,19,32)(9,17,15,24,27,31,26,30);(1,5,7,10,8,21,6,25)(2,11,13,4,14,22,12,23)(3,17,18,24,20,31,19,30)(9,16,15,28,27,29,26,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,g^b=gi,f^c=fi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263723);