Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1302 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 214 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14,11,13,15,16)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16);(3,8)(5,7)(10,13)(12,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,6)(5,7)(10,13)(14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238954, #72238963 |
| maximal normal subgroup of | #72238954, #72238963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120822);