Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #37454 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 887 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,4,6,5,8,3,7)(9,10)(11,19,22,30)(12,23)(13,26,28,17)(14,18)(15,21)(16,24,33,40)(20,35,37,29)(25,32)(27,31)(34,39)(36,38);(2,6)(3,4)(7,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,29)(17,30)(18,31)(19,26)(21,28)(23,38)(24,39)(25,37)(32,34)(33,35)(36,40);(9,12,14,25)(10,16,18,33)(11,20,22,37)(13,24,28,40)(15,29,31,35)(17,23,26,32)(19,34,30,38)(21,36,27,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fg,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285449 |
| maximal normal subgroup of | #72285449 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38488);