Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#331 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(2,6,7,8)(9,10,13,20,15,21,31,38,16,22,30,37,14,19,12,18)(11,23,25,39,27,17,24,33,28,34,29,32,26,35,40,36);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,29)(13,24)(14,27)(15,26)(16,28)(18,36)(19,34)(20,32)(21,23)(22,35)(25,31)(30,40)(33,38)(37,39);(1,3,5,4)(2,6,7,8)(9,12,14,30,16,31,15,13)(10,18,19,37,22,38,21,20)(11,24,26,25,28,40,27,29)(17,32,23,33,35,39,34,36)Polycyclic presentation:
2^9|a^2=e,c^2=fg,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32316);