Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1836 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)(5,6,10,16,12,8,11,15)(7,17,21,23,22,18,20,13)(9,24,25,29,27,14,26,30)(19,32,34,28,35,31,33,36);(2,4)(5,7)(6,13)(8,23)(9,19)(10,21)(11,20)(12,22)(14,28)(15,18)(16,17)(24,36)(25,34)(26,33)(27,35)(29,32)(30,31);(5,8)(6,12)(7,18)(9,24)(10,15)(11,16)(13,21)(14,27)(17,22)(19,32)(20,23)(25,29)(26,30)(28,34)(31,35)(33,36);(5,9,10,25,12,27,11,26)(6,14,16,30,8,24,15,29)(7,19,21,34,22,35,20,33)(13,28,17,31,23,36,18,32)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239830, #72252967, #72256876, … |
| maximal normal subgroup of | #72239830, #72252967, #72256876, … |
| maximal quotient of | #72308380 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266219);