Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1015 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)(5,6,11,15)(7,17,20,22)(8,12,23,19)(9,24,27,31)(10,16,29,14)(13,26,32,28)(18,21,34,36)(25,33,35,30);(5,7)(6,8)(9,25)(10,12)(11,20)(13,30)(14,22)(15,23)(16,17)(18,26)(19,29)(21,31)(24,36)(27,35)(28,34)(32,33);(5,8)(6,12,15,19)(7,14,20,16)(9,21)(10,22)(11,23)(13,25,32,35)(17,29)(18,24,34,31)(26,33)(27,36)(28,30);(5,9,11,27)(6,13,15,32)(7,18,20,34)(8,21,23,36)(10,28,29,26)(12,25,19,35)(14,24,16,31)(17,33,22,30)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293578);