Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188035 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2908 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 2908 conjugacy classes of subgroups — too many to draw. Only its 318 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:
(1,2,4,10)(3,9,13,22)(5,7,15,19)(6,8,16,21)(11,18,23,29)(12,17,25,28)(14,20,26,30)(24,27,31,32);(2,7)(3,11)(4,15)(8,17)(14,24)(16,25)(22,29)(30,32);(2,8)(4,16)(7,17)(9,20)(13,26)(15,25)(18,27)(23,31);(1,3,12,24)(2,9,17,27)(4,13,25,31)(5,11,6,14)(7,18,8,20)(10,22,28,32)(15,23,16,26)(19,29,21,30)Polycyclic presentation:
2^9|a^2=g,d^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72266963, #72266965, #72266967, … |
| maximal normal subgroup of | #72266963, #72266965, #72266967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95242);