Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124874 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1560 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7635 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,12,14,23,15,22,19,7)(4,16)(5,9)(6,11)(8,17)(10,20)(13,27,28,32,29,21,30,26)(18,24)(25,31);(1,3)(2,7)(4,13)(5,19)(6,15)(8,21)(9,22)(10,12)(11,23)(14,20)(16,26)(17,30)(18,29)(24,32)(25,27)(28,31);(1,4,5,17,6,18,20,31)(2,8,10,16,11,25,9,24)(3,13,19,30,15,29,14,28)(7,21,12,26,23,27,22,32)Polycyclic presentation:
2^9|a^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32100);