Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140211 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| 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 | |
730 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)(5,6,11,15,12,18,8,25)(7,19,23,32,24,31,21,13)(9,28,27,17,10,14,26,16)(20,34,22,30,29,33,35,36);(1,2,3,4)(5,7,8,21,12,24,11,23)(6,13,15,19,18,32,25,31)(9,20,26,35,10,29,27,22)(14,30,16,33,28,36,17,34);(6,14)(7,20)(8,11)(9,27)(10,26)(15,17)(16,25)(18,28)(19,31)(21,22)(23,35)(24,29)(30,33)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=h,g^2=i,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=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239813, #72239844, … |
| maximal normal subgroup of | #72239802, #72239813, #72239844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55357);