Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102857 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C43 | C23 |
| socle | C23 | #392 |
| prime radical | #23798 | C2 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
651 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 651 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,3)(4,7,6,8)(9,10,13,16)(11,14,12,15)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(2,7)(3,8)(4,6)(10,14)(11,12)(15,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,31)(25,28)(26,30);(1,3)(2,5)(4,8)(6,7)(10,15)(11,12)(14,16)(22,31)(25,32)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72211656, C2×#102858 |
| maximal normal subgroup of | #72211656, C2×#102858 |
| maximal quotient of | #72211626, #72211633, C2×#102858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10054);