Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
493 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,24,38)(12,17)(13,27)(14,19)(15,20)(16,23,32,39)(18,26)(22,37,25,35)(28,33,40,31)(29,36)(30,34);(2,5)(9,11)(10,16)(12,22)(13,28)(14,25)(15,24)(17,31)(18,35)(19,33)(20,32)(21,34)(23,29)(26,39)(27,38)(30,40)(36,37);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,30,29)(16,31,32,33)(18,34,36,27)(21,37,38,35)(23,28,39,40)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29856);