Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,22)(18,26)(20,27)(23,29)(24,28)(25,30)(31,32)(33,35)(34,36);(1,3,4,2)(5,7,10,19,11,20,9,24)(6,12,14,27,16,28,15,17)(8,18,22,33,23,34,25,32)(13,26,29,35,30,36,21,31);(5,7)(6,12)(8,18)(9,19)(10,24)(11,20)(13,26)(14,17)(15,27)(16,28)(21,35)(22,32)(23,34)(25,33)(29,31)(30,36);(5,8,11,23)(6,13,16,30)(7,18,20,34)(9,25,10,22)(12,26,28,36)(14,29,15,21)(17,31,27,35)(19,33,24,32)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273561);