Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150429 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,19,15,17,12,27)(11,21,24,16,26,34,40,36)(14,31,28,18,32,20,30,37)(22,39,23,33,25,38,29,35);(2,6)(3,4)(7,8)(9,11)(10,16)(12,23)(13,29)(14,22)(15,26)(17,36)(18,34)(19,38)(20,33)(21,37)(24,28)(25,32)(27,39)(30,40)(31,35);(1,3,5,4)(2,6,8,7)(10,17)(11,22)(12,13)(14,32)(16,33)(18,37)(21,38)(23,40)(24,29)(25,26)(34,39)(35,36)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfh,d^b=dg,e^b=efgi,f^b=fi,d^c=dhi,e^c=ei,f^c=fi,e^d=ehi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58032);