Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3554 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2979 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)(13,14)(15,23)(16,27)(17,30)(18,21)(19,26)(20,28)(22,33)(24,34)(25,37)(29,38)(31,35)(32,36)(39,41)(40,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,19,21,35)(16,24,29,40)(17,31,33,26)(20,34,36,43)(22,37,30,23)(27,39,38,44)(28,42,32,41);(6,9)(10,12)(13,16,18,29)(14,20,21,36)(15,24,25,40)(17,32,33,28)(19,34,35,43)(22,38,30,27)(23,39,37,44)(26,42,31,41);(5,7)(6,10)(8,11)(9,12)(14,21)(15,25)(20,36)(22,30)(24,40)(26,31)(27,38)(41,42);(7,11)(10,12)(15,25)(19,35)(23,37)(24,40)(26,31)(34,43)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449490);