Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133080 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 991 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 32 points:
(1,2,5,12,6,13,16,25)(3,14,17,26,15,32,19,8)(4,10,18,9,20,28,22,29)(7,11,23,30,24,31,21,27);(2,8)(3,15)(4,19)(5,21)(9,12)(10,11)(13,26)(14,30)(16,23)(17,20)(18,22)(25,29)(27,32)(28,31);(1,3)(2,9)(4,16)(5,20)(6,15)(7,18)(8,25)(10,27)(11,14)(12,26)(13,29)(17,23)(19,21)(22,24)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72238366, #72238367, #72238370, … |
| maximal normal subgroup of | #72238366, #72238367, #72238370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40282);