Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1222 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(6,9)(14,15)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311007, #72338954, #72338972, … |
| maximal normal subgroup of | #72311007, #72338954, #72338972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121038);