Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
859 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)(7,11)(8,19)(9,14)(10,22)(12,21)(13,15)(16,32)(17,26)(18,28)(20,30)(23,27)(24,31)(25,33)(29,35)(34,36);(5,7,15,17,9,18,10,23)(6,11,22,27,14,28,13,26)(8,16,24,33,20,34,21,35)(12,29,30,36,31,25,19,32);(5,8,9,20)(6,12,14,31)(7,16)(10,24,15,21)(11,25)(13,30,22,19)(17,35)(18,34)(23,33)(26,32)(27,36)(28,29);(6,13)(10,15)(11,26)(12,30)(14,22)(17,23)(19,31)(21,24)(25,32)(27,28)(29,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=def,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,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291179);