Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131212 |
| 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 | #2828 | C4 |
| 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 | |
540 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 540 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,20,15,21,12,18)(11,22,23,16,25,31,29,32)(13,17,26,34,28,19,27,33)(24,36,39,30,40,37,38,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,23)(13,24)(14,29)(15,25)(17,30)(18,22)(19,35)(20,31)(21,32)(26,38)(27,39)(28,40)(33,36)(34,37);(10,17,21,19)(12,14)(16,30,32,35)(18,33,20,34)(22,36,31,37)(23,29)(26,27)(38,39)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38416);