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 | C25 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3928 conjugacy classes of subgroups; only the 416 normal subgroups are shown.
This group has 3928 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)(3,7,13,19)(4,8)(5,9)(6,10)(11,17,24,29)(12,18,25,30)(14,26)(15,21)(16,22)(20,27)(23,31,32,28);(1,3)(2,7,10,19)(4,11)(5,12)(6,13)(8,17,21,29)(9,18,22,30)(14,23)(15,24)(16,25)(20,28,26,31)(27,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,20)(10,21)(12,23)(13,24)(16,27)(18,28)(19,29)(22,26)(25,32)(30,31);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,20)(10,22)(11,23)(13,25)(15,27)(17,28)(19,30)(21,26)(24,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=eh,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 | #72234172, #72237754, #72237760, … |
| maximal normal subgroup of | #72234172, #72237754, #72237760, … |
| maximal quotient of | #72234150, #72258512, #72261568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299915);