Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1263 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)(5,6,10,16,9,14,26,15)(7,17,21,31,19,11,20,13)(8,22,24,12,25,29,34,30)(18,33,23,27,36,28,35,32);(5,7,9,19)(6,11,14,17)(8,23)(10,21,26,20)(12,28)(13,15,31,16)(18,34)(22,27)(24,36)(25,35)(29,32)(30,33);(5,8,10,24,9,25,26,34)(6,12,15,22,14,30,16,29)(7,18,20,35,19,36,21,23)(11,27,13,28,17,32,31,33);(2,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,16)(12,28)(14,31)(15,17)(21,26)(22,32)(23,34)(24,35)(25,36)(27,29)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341029);