Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,14,10,26)(7,16,19,28,20,12,18,29)(8,21,23,31,25,32,24,13)(17,34,35,27,36,30,22,33);(1,2)(3,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,16)(15,28)(17,24)(21,27)(23,36)(25,35)(26,29)(31,34)(32,33);(2,4)(5,8,11,25)(6,13,14,31)(7,17,20,36)(9,24,10,23)(12,27,16,33)(15,32,26,21)(18,35,19,22)(28,30,29,34);(6,14)(7,18)(9,10)(12,28)(16,29)(17,35)(19,20)(21,32)(22,36)(23,24)(27,34)(30,33)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383495);