Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2498 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 2498 conjugacy classes of subgroups — too many to draw. Only its 346 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:
(1,2)(3,4)(5,6,9,14,10,11,16,25)(7,12,18,27,15,24,22,29)(8,19,21,30,20,13,23,28)(17,26,32,35,31,34,33,36);(2,4)(6,11)(7,15)(9,16)(12,24)(13,19)(17,31)(21,23)(26,34);(5,7,16,18,10,15,9,22)(6,12,14,29,11,24,25,27)(8,17,21,33,20,31,23,32)(13,26,30,35,19,34,28,36);(2,4)(5,8)(6,13)(7,17)(9,23)(10,20)(11,19)(12,26)(14,30)(15,31)(16,21)(18,33)(22,32)(24,34)(25,28)(27,36)(29,35)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341298, #72341301, #72341366, … |
| maximal normal subgroup of | #72341298, #72341301, #72341366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281206);