Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13469 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2065 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,19,24,13)(8,25,22,35)(9,29)(10,15)(11,18)(12,21,33,23)(14,30)(16,26)(17,20)(27,32,28,34)(31,36);(1,2,3,4)(5,7,9,21,10,22,30,28,11,24,31,23,20,8,26,27)(6,12,14,19,17,34,29,25,18,33,16,13,15,32,36,35);(5,8)(6,13)(7,20)(9,21)(10,24)(11,22)(12,14)(15,35)(16,33)(17,25)(18,19)(23,31)(26,28)(27,30)(29,32)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250132, #72250139, #72250188, … |
| maximal normal subgroup of | #72250132, #72250139, #72250188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55559);