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 | #38916 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 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)(21,22,25,26)(23,27,32,34)(24,29,33,28)(30,31,36,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23)(22,27)(24,31)(25,32)(26,34)(28,30)(29,36)(33,35);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24,25,33)(22,28,26,29)(23,30,32,36)(27,35,34,31);(6,13)(11,15)(12,14)(19,20)(21,24)(22,29)(23,31)(25,33)(26,28)(27,36)(30,34)(32,35);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,22)(24,28)(25,26)(27,34)(29,33)(31,35)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6581564);