Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501527 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
707 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 707 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:
(5,6,9,12,18,20,7,15,8,14,17,11,10,13,16,19);(1,2,3,4)(7,16)(9,17)(11,12)(15,19);(2,4)(5,6)(7,11)(8,14)(9,19)(10,20)(12,16)(13,18)(15,17);(5,7,10,9,8,16,18,17)(6,11,20,19,14,12,13,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72224852 |
| maximal normal subgroup of | #72224852 |
| maximal quotient of | #72224850 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413775);