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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 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)(5,6,10,16,12,9,11,15)(7,13,20,29,21,18,19,30)(8,22,26,14,25,32,27,24)(17,31,23,33,34,36,35,28);(5,7)(6,13)(8,23,25,35)(9,18)(10,20)(11,19)(12,21)(14,31,24,36)(15,30)(16,29)(17,26,34,27)(22,28,32,33);(5,8,10,26,12,25,11,27)(6,14,15,22,9,24,16,32)(7,17,19,35,21,34,20,23)(13,28,29,31,18,33,30,36);(2,4)(5,9,12,6)(7,18,21,13)(8,24,25,14)(10,16,11,15)(17,33,34,28)(19,30,20,29)(22,26,32,27)(23,31,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,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, 341030);