Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#3271 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,24,19,22,17,23)(15,20,28,35,26,36,27,34)(16,29,30,37,31,38,32,21)(25,39,41,43,40,44,42,33);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,22,36)(16,25,31,40)(17,27,18,28)(21,33,37,43)(23,34,24,35)(29,39,38,44)(30,41,32,42);(6,9)(10,12)(13,16,19,31)(14,21,22,37)(15,25,26,40)(17,30,18,32)(20,33,36,43)(23,29,24,38)(27,41,28,42)(34,39,35,44);(5,7)(6,10)(8,11)(9,12)(14,22)(15,26)(21,37)(23,24)(25,40)(27,28)(29,38)(41,42);(7,11)(10,12)(13,17,19,18)(14,23,22,24)(15,27,26,28)(16,30,31,32)(20,34,36,35)(21,29,37,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452214);