Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,24)(18,21)(19,33)(23,30)(25,29)(26,28)(27,35)(31,34)(32,36);(1,2)(3,4)(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,20)(17,21)(19,32)(22,24)(23,29)(25,30)(26,28)(27,34)(31,35)(33,36);(1,3,4,2)(5,7)(6,12)(8,19)(9,26)(10,20)(11,21)(13,27)(14,18)(15,17)(16,28)(22,31)(23,36)(24,34)(25,35)(29,32)(30,33);(5,8,9,23,11,25,10,24)(6,13,14,29,16,30,15,22)(7,19,26,36,21,35,20,34)(12,27,18,32,28,33,17,31)Polycyclic presentation:
2^9|c^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319234);