Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2001 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)(5,6,9,10)(7,11,8,12)(13,14,18,16)(15,23,27,20)(17,21,33,28)(19,22,35,29)(24,26,40,36)(25,38,30,37)(31,32,39,34)(41,43,44,42);(6,10)(11,12)(13,15,18,27)(14,20,16,23)(17,30,33,25)(19,36,35,26)(21,38,28,37)(22,40,29,24)(31,44,39,41)(32,43,34,42);(2,4)(13,16,18,14)(15,24,27,40)(17,31,33,39)(19,22,35,29)(20,36,23,26)(21,32,28,34)(25,38,30,37)(41,42,44,43);(5,7,9,8)(6,11,10,12)(13,17)(14,21)(15,25)(16,28)(18,33)(19,32)(20,37)(22,39)(23,38)(24,41)(26,43)(27,30)(29,31)(34,35)(36,42)(40,44);(5,8,9,7)(6,12,10,11)(15,26)(17,32)(19,35)(20,24)(21,39)(22,29)(23,40)(25,30)(27,36)(28,31)(33,34)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485398);