Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1343 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| socle | C2 | C2×#1343 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,11,7,12,4,9)(3,13,16,30,17,26,14,29)(5,19,20,25,21,10,18,27)(8,22,23,15,24,31,28,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,23)(10,26)(11,28)(12,24)(13,19)(18,31)(20,22)(21,32)(25,30)(27,29);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,21,20)(8,23,24,28)(10,25,19,27)(13,29,26,30)(15,31,32,22)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,e^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,h^b=hi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57365);