Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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×#3321 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2014 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,36)(26,28)(27,40)(29,37)(30,38)(31,35)(32,34)(33,39)(41,44)(42,43);(6,9)(7,11)(13,15)(14,20)(16,30)(17,26)(18,27)(19,29)(21,38)(22,28)(23,37)(24,40)(25,32)(31,44)(33,43)(34,36)(35,41)(39,42);(2,4)(13,16,19,33)(14,21,23,39)(15,25,29,41)(17,34,24,31)(18,35,22,32)(20,36,37,44)(26,42,40,38)(27,30,28,43);(6,9)(10,12)(13,17,19,24)(14,22,23,18)(15,26,29,40)(16,31,33,34)(20,28,37,27)(21,35,39,32)(25,38,41,42)(30,44,43,36);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(18,22)(20,26)(21,39)(25,30)(28,29)(32,35)(36,42)(37,40)(38,44)(41,43)Polycyclic presentation:
2^9|c^2=i,d^2=i,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,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579988);