Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12696 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #60801 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 683 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,10,6,11,21,28)(3,12,15,27,14,24,29,32)(4,8,17,25,18,9,20,26)(7,19,23,16,22,30,31,13);(1,3)(2,7)(4,16)(5,19)(6,14)(8,12)(9,24)(10,27)(11,22)(13,18)(15,17)(20,29)(21,30)(23,26)(25,31)(28,32);(1,4,6,18)(2,8,11,9)(3,13,14,16)(5,17,21,20)(7,12,22,24)(10,25,28,26)(15,19,29,30)(23,27,31,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59302);