Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C23 | #441 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,6,8,3)(9,10,12,19,13,17,30,36,15,21,29,18,14,20,31,35)(11,22,25,34,26,38,28,32,27,39,24,33,23,37,40,16);(2,6)(3,7)(9,11,13,23,15,27,14,26)(10,16,17,33,21,32,20,34)(12,28,30,25,29,40,31,24)(18,37,35,39,19,38,36,22);(1,3,8,6,4,7,5,2)(10,17,21,20)(11,23,27,26)(12,29)(16,32)(18,36,19,35)(24,40,25,28)(30,31)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,c^b=ch,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320774, #72320784, #72320793, … |
| maximal normal subgroup of | #72320774, #72320784, #72320793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36374);