Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,39)(27,34)(28,36)(29,37)(30,38)(32,35)(33,42)(40,44)(41,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,23,35)(16,25,28,40)(17,26,18,32)(21,33,36,43)(22,34,31,24)(29,41,30,42)(37,39,38,44);(1,2,3,4)(6,9)(7,11)(16,28)(21,36)(25,40)(29,30)(33,43)(37,38)(39,44)(41,42);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,30)(18,29)(19,28)(20,33)(22,38)(23,36)(24,39)(26,42)(27,40)(31,37)(32,41)(34,44)(35,43);(6,9)(10,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,28,30)(20,34,35,24)(21,37,36,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426999);