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×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1202 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)(5,6,10,16,12,17,11,15)(7,18,21,13,22,32,23,31)(8,9,26,28,14,30,25,29)(19,27,24,36,35,34,33,20);(5,7,10,21,12,22,11,23)(6,13,15,18,17,31,16,32)(8,24,25,19,14,33,26,35)(9,27,28,36,30,34,29,20);(5,8)(6,9)(7,19,22,35)(10,26)(11,25)(12,14)(13,20,31,36)(15,29)(16,28)(17,30)(18,34,32,27)(21,24,23,33);(2,4)(5,9,12,30)(6,14,17,8)(7,20,22,36)(10,29,11,28)(13,19,31,35)(15,25,16,26)(18,24,32,33)(21,34,23,27)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=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, 341114);