Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501052 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1116 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)(5,6,8,13,21,31,35,30,10,15,22,29,9,14,20,12)(7,11,17,24,18,25,34,36,19,27,33,23,16,26,32,28);(2,4)(5,7)(6,11)(8,17)(9,16)(10,19)(12,28)(13,24)(14,26)(15,27)(18,21)(20,32)(22,33)(23,29)(25,31)(30,36)(34,35);(6,12)(7,16)(8,20)(9,21)(11,23)(13,14)(15,30)(17,33)(18,19)(22,35)(24,27)(25,36)(26,28)(29,31);(2,4)Polycyclic presentation:
2^9|a^2=f,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408248);