Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,17,12,18,9,15)(7,19,21,13,22,32,8,23)(10,26,28,30,29,35,25,16)(14,20,33,36,34,27,31,24);(1,2)(3,4)(5,7)(6,13)(8,11)(9,21)(10,27)(12,22)(14,26)(15,32)(16,33)(17,19)(18,23)(20,29)(24,25)(28,36)(30,31)(34,35);(1,3,4,2)(5,8,9,22,12,21,11,7)(6,14,15,31,18,34,17,33)(10,20,25,24,29,27,28,36)(13,30,19,26,23,16,32,35)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250068, #72250082, #72250104, … |
| maximal normal subgroup of | #72250068, #72250082, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55822);