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×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,18,25,17)(7,19,21,35,23,34,27,13)(8,24,10,14,26,16,30,36)(12,28,31,29,33,22,32,20);(5,7,10,28,9,21,26,29,11,23,30,22,25,27,8,20)(6,12,24,13,17,32,36,34,18,33,16,35,15,31,14,19);(2,4)(5,7)(6,13)(8,22)(9,27)(10,29)(11,23)(12,16)(14,31)(15,34)(17,19)(18,35)(20,30)(21,25)(24,33)(26,28)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250029, #72250050, #72250104, … |
| maximal normal subgroup of | #72250029, #72250050, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55627);