Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1057 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)(6,11)(7,15)(8,20)(9,24)(13,21)(14,25)(16,26)(17,31)(18,19)(27,30)(32,36)(34,35);(5,6,10,14)(7,16,19,26)(8,21,20,27)(9,25,24,11)(12,18,29,15)(13,23,30,22)(17,33,35,28)(31,32,34,36);(1,3,4,2)(5,7)(6,12)(8,17)(9,18)(10,19)(11,26)(13,28)(14,29)(15,24)(16,25)(20,35)(21,32)(22,31)(23,34)(27,36)(30,33);(5,8)(6,13)(7,17)(9,23)(10,20)(11,27)(12,28)(14,30)(15,31)(16,32)(18,34)(19,35)(21,25)(22,24)(26,36)(29,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295389);