Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,15)(8,9)(12,16)(13,20)(14,19)(17,18);(1,2)(3,4)(6,12,13,14)(11,19,20,16);(2,4)(5,6)(7,16)(8,14)(9,12)(10,13)(11,15)(17,19)(18,20);(7,17)(11,20)(15,18)(16,19)Polycyclic presentation:
2^9|b^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214893, #72226642, #72226644, … |
| maximal normal subgroup of | #72214893, #72226642, #72226644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407142);