Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130074 |
| 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×Q8 | C23 |
| prime radical | C2×#127 | D4 |
| 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 | |
653 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 653 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,4,6,5,8,3,7)(9,10)(11,20,23,35)(12,17,25,32)(13,18)(14,19)(15,21)(16,24,31,26)(22,28,39,40)(27,34)(29,36)(30,37)(33,38);(2,6)(3,4)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,28)(17,20)(18,33)(19,30)(24,36)(25,37)(26,38)(27,39)(29,34)(31,40)(32,35);(9,12,14,25)(10,16,19,31)(11,21,23,37)(13,26,27,24)(15,28,30,40)(17,18,32,34)(20,33,35,29)(22,38,39,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,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, 37410);