Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124937 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1533 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,7,8,5)(4,6)(9,10,13,18,15,19,27,35,14,20,28,36,12,17,25,33)(11,16,23,30,21,29,37,34,24,32,38,40,22,31,26,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,26)(14,24)(15,22)(17,29)(18,34)(19,31)(20,32)(23,25)(27,38)(28,37)(30,35)(33,40)(36,39);(2,6)(5,7)(9,12,14,15)(10,17,20,19)(11,21,24,22)(13,27,28,25)(16,29,32,31)(18,35,36,33)(23,26,38,37)(30,39,40,34)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32135);