Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150397 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| 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 ◁ C23 ◁ C2×C42 ◁ #2327 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 54 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)(9,10,12,18,14,19,25,34)(11,16,22,31,21,32,36,40)(13,15,23,29,26,17,24,33)(20,30,35,39,27,28,37,38);(2,6)(3,4)(7,8)(10,15)(11,20)(12,23)(13,26)(17,19)(21,27)(22,36)(24,25)(28,30)(29,33)(31,38)(39,40);(1,3,5,4)(2,7,8,6)(9,11)(10,16)(12,22)(13,27)(14,21)(15,28)(17,30)(18,31)(19,32)(20,26)(23,37)(24,35)(25,36)(29,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=e,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,d^c=dh,f^c=fi,e^d=ehi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72292967, #72292970, #72293122, … |
| maximal normal subgroup of | #72292967, #72292970, #72293122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57677);