Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | #1459 |
| prime radical | #392 | C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5)(4,6)(7,8)(9,10,12,17,13,18,25,32,14,15,24,27,26,28,34,38)(11,19,20,31,21,33,35,40,22,16,23,29,36,30,37,39);(2,5)(3,8)(6,7)(10,15)(16,19)(18,28)(30,33);(2,6)(5,7)(9,11,13,21,14,22,26,36)(10,16,18,30,15,19,28,33)(12,23,25,37,24,20,34,35)(17,31,32,40,27,29,38,39)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,d^b=dg,e^b=efg,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239990, #72239992, #72239994, … |
| maximal normal subgroup of | #72239990, #72239992, #72239994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31585);