Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499263 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
784 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,7,14,19,13,20,15,10,16,18,12,9,17,8,11);(1,2)(3,4)(6,11)(7,8)(9,19)(12,13)(14,17)(15,16)(18,20);(2,4)(6,12)(7,8)(9,19)(11,17)(13,15)(14,16)(18,20);(6,13,16,17)(7,18)(8,20)(11,12,15,14)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#499656, #72217802, #72217807, … |
| maximal normal subgroup of | C2×#499656, #72217802, #72217807, … |
| maximal quotient of | #72216645, #72216658, C2×#499656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406852);