Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,9,14,10,23)(7,12,24,27,18,28,17,31)(8,19,21,13,22,32,25,30)(16,33,35,26,20,36,34,29);(1,2)(3,4)(5,7)(6,12)(8,20)(9,18)(10,24)(11,17)(13,29)(14,28)(15,31)(16,22)(19,33)(21,35)(23,27)(25,34)(26,30)(32,36);(2,4)(5,8,11,21,9,22,10,25)(6,13,15,32,14,30,23,19)(7,16,24,35,18,20,17,34)(12,26,27,36,28,29,31,33);(6,14)(7,17)(10,11)(12,27)(16,34)(18,24)(19,32)(20,35)(21,25)(26,33)(28,31)(29,36)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384518);