Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(22,27)(23,31)(30,34)(32,33);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,22,25,27)(23,29,31,36)(24,30,26,34)(28,33,35,32);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23,25,31)(22,28,27,35)(24,32,26,33)(29,30,36,34);(6,13)(11,15)(12,14)(19,20)(21,23)(22,29)(24,33)(25,31)(26,32)(27,36)(28,34)(30,35);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,24)(22,27)(23,32)(25,26)(28,35)(31,33)Polycyclic presentation:
2^9|b^2=i,c^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581521);