Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,27)(25,38)(26,34)(28,39)(29,35)(30,36)(31,37)(33,40)(41,43)(42,44);(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,28,17,26)(14,20,22,24,23,34,32,39)(16,25,29,40,30,41,31,42)(21,33,37,38,36,44,35,43);(2,4)(7,11)(9,12)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,33)(22,37)(23,36)(24,38)(26,42)(27,40)(28,41)(32,35)(34,44)(39,43);(5,7)(6,9)(8,11)(10,12)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,29,30,31)(20,24,34,39)(21,35,36,37)(25,40,41,42)(33,43,44,38);(7,11)(9,12)(16,30)(21,36)(25,41)(29,31)(33,44)(35,37)(38,43)(40,42)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498855);