Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392719 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3687 conjugacy classes of subgroups; only the 416 normal subgroups are shown.
This group has 3687 conjugacy classes of subgroups — too many to draw. Only its 416 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:
(1,2,6,11,7,10,19,3)(4,8,16,22,15,21,29,12)(5,9,18,24,17,23,30,13)(14,26,28,25,27,20,31,32);(1,3,6,10,7,11,19,2)(4,12,16,21,15,22,29,8)(5,13,18,23,17,24,30,9)(14,25,28,26,27,32,31,20);(1,4)(2,8)(3,12)(5,14)(6,16)(7,15)(9,20)(10,21)(11,22)(13,25)(17,27)(18,28)(19,29)(23,26)(24,32)(30,31);(1,5)(2,9)(3,13)(4,14)(6,18)(7,17)(8,20)(10,23)(11,24)(12,25)(15,27)(16,28)(19,30)(21,26)(22,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi,g^c=gi,f^d=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72215925, #72215931, … |
| maximal normal subgroup of | #72215906, #72215925, #72215931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299916);