Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14609 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 196 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:
(1,2,6,11,7,10,23,4)(3,12,16,17,15,8,24,13)(5,19,21,26,20,9,22,18)(14,27,30,28,29,25,31,32);(1,3)(2,8,10,12)(4,17,11,13)(5,14,20,29)(6,16)(7,15)(9,25)(18,32)(19,27)(21,30,22,31)(23,24)(26,28);(1,4,6,10,7,11,23,2)(3,13,16,8,15,17,24,12)(5,18,22,19,20,26,21,9)(14,28,31,25,29,32,30,27);(1,5)(2,9)(3,14,15,29)(4,18)(6,22)(7,20)(8,25,12,27)(10,19)(11,26)(13,28,17,32)(16,31,24,30)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284788, #72284899, #72284925, … |
| maximal normal subgroup of | #72284788, #72284899, #72284925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298292);