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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,27)(17,22)(18,31)(19,25)(20,29)(23,30)(24,39)(26,34)(28,38)(32,36)(33,43)(35,41)(37,42)(40,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,33)(21,22)(23,34)(27,44)(28,40)(29,41)(30,42)(31,37)(32,43)(35,36)(38,39);(2,4)(13,16,17,28)(14,20,22,36)(15,24,25,40)(18,32,30,29)(19,33,21,35)(23,38,31,27)(26,43,42,41)(34,39,37,44);(2,4)(13,15)(14,21)(16,24)(17,25)(18,26)(19,22)(20,35)(23,37)(27,39)(28,40)(29,41)(30,42)(31,34)(32,43)(33,36)(38,44);(14,22)(15,25)(20,36)(23,31)(24,40)(26,42)(27,38)(41,43)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345342);