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 | #2011 | C4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 343 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)(9,10,12,17,14,19,26,36)(11,20,22,35,24,32,38,40)(13,16,29,34,28,18,27,37)(15,30,31,23,33,25,39,21);(2,6)(3,4)(7,8)(9,11)(10,15)(12,25)(13,21)(14,24)(16,32)(17,35)(18,20)(19,33)(22,29)(23,28)(26,30)(27,38)(31,34)(36,40)(37,39);(1,3,5,4)(2,7,8,6)(11,21)(13,28)(15,20)(16,18)(22,30)(23,24)(25,38)(27,29)(31,35)(32,33)(34,37)(39,40)Polycyclic presentation:
2^9|a^2=e,c^2=gh,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 | #72292954, #72292955, #72292960, … |
| maximal normal subgroup of | #72292954, #72292955, #72292960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57676);