Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #47800 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 711 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:
(1,2,4,6,5,8,3,7)(9,10)(11,20,23,35)(12,24,27,17)(13,16)(14,19)(15,21)(18,29)(22,37,28,38)(25,33)(26,34,30,36)(31,39)(32,40);(2,6)(3,4)(7,8)(9,11,14,23)(10,15,19,33)(12,25,27,21)(13,28,29,22)(16,31,18,32)(17,35,24,20)(26,39,30,40)(34,37,36,38);(10,16)(13,29)(15,31)(17,36)(18,19)(20,37)(22,28)(24,34)(26,30)(32,33)(35,38)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72240240, #72240242, #72240244, … |
| maximal normal subgroup of | #72240240, #72240242, #72240244, … |
| maximal quotient of | #72240310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37478);