Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1340 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)(5,6,11,15,12,8,10,23)(7,16,18,13,20,29,19,21)(9,22,26,32,27,14,25,31)(17,30,24,28,35,33,34,36);(5,7)(6,13)(8,21)(9,24)(10,19)(11,18)(12,20)(14,30)(15,29)(16,23)(17,25)(22,33)(26,35)(27,34)(28,31)(32,36);(2,4)(5,8)(6,12)(7,13,20,21)(9,22)(10,23)(11,15)(14,27)(16,19,29,18)(17,28,35,36)(24,30,34,33)(25,32)(26,31);(2,4)(5,9,12,27)(6,14,8,22)(7,17,20,35)(10,26,11,25)(13,28,21,36)(15,32,23,31)(16,33,29,30)(18,34,19,24)Polycyclic presentation:
2^9|a^2=h,c^2=egh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297612);