Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
734 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)(5,6,11,16)(7,17,22,29)(8,24,25,14)(9,13,27,34)(10,28,26,15)(12,23,32,19)(18,33,20,30)(21,31,36,35);(5,7,8,19,11,22,25,23)(6,12,14,17,16,32,24,29)(9,21,26,18,27,36,10,20)(13,33,28,31,34,30,15,35);(6,13)(7,18)(8,25)(9,26)(10,27)(12,29)(14,15)(16,34)(17,32)(19,21)(20,22)(23,36)(24,28)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250368, #72250374, #72250383, … |
| maximal normal subgroup of | #72250368, #72250374, #72250383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55350);