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 | #35676 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2468 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 2468 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)(5,6,10,15,11,14,28,8)(7,16,19,17,20,12,26,21)(9,23,25,30,24,13,27,22)(18,29,34,32,33,31,35,36);(2,4)(5,7,11,20)(6,12,14,16)(8,21,15,17)(9,18,24,33)(10,26,28,19)(13,29,23,31)(22,36,30,32)(25,35,27,34);(5,8,10,14,11,15,28,6)(7,17,19,16,20,21,26,12)(9,22,27,23,24,30,25,13)(18,32,35,31,33,36,34,29);(2,4)(5,9)(6,13)(7,18)(8,22)(10,27)(11,24)(12,29)(14,23)(15,30)(16,31)(17,32)(19,35)(20,33)(21,36)(25,28)(26,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,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 | #72317994, #72318099, #72318241, … |
| maximal normal subgroup of | #72317994, #72318099, #72318241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281207);