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 | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2648 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 2648 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)(7,15,18,11)(8,12)(9,22,23,13)(10,14)(16,28,19,24)(17,25)(20,26,33,34)(21,27)(29,31)(30,36)(32,35);(2,4)(5,7)(6,11)(8,19)(9,17)(10,18)(12,24)(13,25)(14,15)(16,21)(20,32)(22,29)(23,31)(26,35)(27,28)(30,33)(34,36);(5,8,10,21)(6,12,14,27)(7,16)(9,20,23,33)(11,24)(13,26,22,34)(15,28)(17,30)(18,19)(25,35)(29,36)(31,32);(2,4)(5,9)(6,13)(7,17)(8,20)(10,23)(11,25)(12,26)(14,22)(15,29)(16,30)(18,31)(19,32)(21,33)(24,35)(27,34)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252227, #72283508, #72283616, … |
| maximal normal subgroup of | #72252227, #72283508, #72283616, … |
| maximal quotient of | #72252222 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281201);