Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130089 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C23 | #379 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,8,6,3,4,7)(9,10)(11,20)(12,25,27,17)(13,18)(14,19)(15,22,32,21)(16,26,35,28)(23,30)(24,37)(29,34)(31,39,36,38)(33,40);(1,3,6,2)(4,7,5,8)(9,11)(10,15)(12,22)(13,23)(14,24)(16,33)(17,20)(18,36)(19,32)(21,27)(25,37)(26,39)(28,38)(29,40)(30,35)(31,34);(9,12,14,27)(10,16,19,35)(11,21,24,22)(13,28,29,26)(15,30,32,33)(17,34,25,18)(20,36,37,31)(23,39,40,38)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37507);