Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,13,23,33,9,25,11,14,24,32,10,16,22,15)(7,17,18,34,27,30,36,29,21,35,26,31,20,28,19,12);(1,2,3,4)(5,7)(6,12)(8,19)(9,26)(10,27)(11,21)(13,28)(14,29)(15,17)(16,34)(18,22)(20,23)(24,36)(25,35)(30,32)(31,33);(6,13)(8,22)(9,24)(10,23)(12,28)(14,32)(15,33)(16,25)(17,31)(18,19)(20,27)(26,36)(29,30)(34,35);(6,14)(12,29)(13,32)(15,25)(16,33)(17,35)(28,30)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249880, #72249887, #72249915, … |
| maximal normal subgroup of | #72249880, #72249887, #72249915, … |
| maximal quotient of | #72265664, #72280040, C2×#494518 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401714);