Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1084 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)(5,6,10,12,11,16,9,15)(7,13,19,28,20,26,17,25)(8,21,22,29,23,30,24,14)(18,32,34,36,33,35,31,27);(6,12)(7,17)(8,22)(9,10)(13,26)(14,29)(15,16)(18,33)(19,20)(23,24)(27,32)(35,36);(5,7)(6,13,16,26)(8,18)(9,19)(10,17)(11,20)(12,25,15,28)(14,27,29,36)(21,35,30,32)(22,31)(23,33)(24,34);(5,8,11,23)(6,14)(7,18,20,33)(9,24,10,22)(12,21)(13,27)(15,30)(16,29)(17,31,19,34)(25,35)(26,36)(28,32)Polycyclic presentation:
2^9|a^2=h,c^2=efgi,d^2=efg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,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, 295276);