Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2009 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)(7,18)(8,13)(9,16)(10,15)(11,17)(12,20)(14,24)(19,34)(21,30)(22,29)(23,33)(25,32)(26,31)(27,36)(28,35);(5,7)(6,12)(8,23)(9,22)(10,20)(11,21)(13,28)(14,33)(15,30)(16,18)(17,29)(19,24)(25,36)(26,35)(27,31)(32,34);(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,27)(14,16)(15,31)(17,32)(18,28)(20,23)(21,35)(22,36)(29,33)(30,34);(5,8)(6,14)(7,19)(9,26)(10,24)(11,25)(12,28)(13,16)(15,32)(17,31)(18,27)(20,36)(21,35)(22,23)(29,34)(30,33)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290365);