Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,17,10,16,9,15)(7,18,21,12,22,29,20,14)(8,23,25,33,26,13,27,32)(19,30,34,28,35,36,24,31);(1,2,4,3)(5,7)(6,12)(8,24,26,34)(9,21)(10,22)(11,20)(13,30,23,36)(14,16)(15,29)(17,18)(19,25,35,27)(28,32,31,33);(5,8)(6,13)(7,19)(9,27)(10,26)(11,25)(12,28)(14,31)(15,33)(16,23)(17,32)(18,30)(20,24)(21,34)(22,35)(29,36);(1,3,4,2)(5,7)(6,14)(8,19)(9,21)(10,22)(11,20)(12,16)(13,31)(15,18)(17,29)(23,28)(24,25)(26,35)(27,34)(30,33)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279909);