Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2179 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)(5,6,8,9)(7,10,11,12)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,28)(26,40)(27,38)(29,42)(30,44)(31,39)(32,43)(33,37)(34,41)(35,36);(6,9)(10,12)(13,15)(14,20)(16,30)(17,32)(18,28)(19,35)(21,37)(22,39)(23,25)(24,42)(26,31)(27,33)(29,34)(36,41)(38,44)(40,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,21,23,38)(15,26,28,17)(19,32,34,31)(20,22,25,40)(24,43,41,39)(29,33,35,30)(36,44,42,37);(13,17,18,26)(14,22,23,40)(15,27,28,16)(19,30,34,33)(20,21,25,38)(24,44,41,37)(29,31,35,32)(36,43,42,39);(14,23)(15,28)(16,27)(22,40)(24,41)(29,35)(31,32)(37,44)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345684);