Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,11,17)(7,18,23,32)(8,9,14,29)(10,16,30,15)(12,22,33,21)(13,25,28,26)(19,27,35,31)(20,24,36,34);(5,7)(6,12)(8,24)(9,27)(10,22)(11,23)(13,20)(14,34)(15,32)(16,18)(17,33)(19,26)(21,30)(25,35)(28,36)(29,31);(5,8)(6,13)(7,19)(9,16)(10,25)(11,14)(12,31)(15,29)(17,28)(18,20)(21,34)(22,24)(23,35)(26,30)(27,33)(32,36);(5,9,10,13,11,29,30,28)(6,14,15,26,17,8,16,25)(7,20,21,27,23,36,22,31)(12,19,18,34,33,35,32,24)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,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, 290753);