Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #407055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14946 |
| 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 | |
1175 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 1175 conjugacy classes of subgroups — too many to draw. Only its 224 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:
(1,2,7,11)(3,8,17,13)(4,12,10,6)(5,9)(14,27)(15,24,16,18)(19,22)(20,25)(21,26)(23,28)(29,32)(30,31);(1,3,12,15,7,17,6,16)(2,8,10,24,11,13,4,18)(5,14,21,30,20,28,22,29)(9,23,26,32,25,27,19,31);(1,4,6,2,7,10,12,11)(3,13,15,24,17,8,16,18)(5,19,22,9,20,26,21,25)(14,27,30,32,28,23,29,31);(1,5)(2,9)(3,14)(4,19)(6,22)(7,20)(8,23)(10,26)(11,25)(12,21)(13,27)(15,30)(16,29)(17,28)(18,31)(24,32)Polycyclic presentation:
2^9|a^2=ei,b^2=efh,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314252);