Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214040 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #37302 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1024 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14)(18,23)(20,27)(22,28)(26,31);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121236);