Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136733 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| 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 | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,17)(7,19)(8,26,16,9)(10,29,30,15)(12,28,14,18)(13,32)(20,23)(21,35)(22,34)(24,33)(25,27)(31,36);(1,2)(3,4)(5,7)(6,13)(8,27)(9,23)(10,21)(11,25)(12,31)(14,24)(15,33)(16,19)(17,35)(18,20)(22,26)(28,34)(29,36)(30,32);(2,4)(5,8,9,17,12,29,30,28,11,16,26,6,14,15,10,18)(7,20,21,33,24,13,22,19,25,34,32,36,31,35,23,27)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250344, #72250370, #72250382, … |
| maximal normal subgroup of | #72250344, #72250370, #72250382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55744);