Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33638 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,19)(7,20)(8,22)(9,16,13,32)(10,18,31,36)(11,17,28,35)(14,34)(15,30)(21,27)(23,25)(24,26)(29,33);(2,4)(5,7,9,15,12,24,13,21)(6,14,32,33,19,8,16,25)(10,29,28,34,31,23,11,22)(17,27,36,26,35,30,18,20);(5,8)(6,15)(7,16)(9,25)(10,30)(11,26)(12,14)(13,33)(17,34)(18,29)(19,21)(20,28)(22,35)(23,36)(24,32)(27,31);(5,9,12,13)(6,16,19,32)(7,21,24,15)(8,25,14,33)(10,28,31,11)(17,18,35,36)(20,27,26,30)(22,23,34,29)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250040, #72250051, #72250077, … |
| maximal normal subgroup of | #72250040, #72250051, #72250077, … |
| maximal quotient of | #72264894 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401521);