Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,27)(26,39)(28,34)(29,36)(30,37)(31,38)(32,35)(33,41)(40,43)(42,44);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,34)(23,25)(24,35)(29,40)(30,41)(31,42)(36,39)(37,44)(38,43);(1,2,3,4)(6,9)(7,11)(13,16,19,31)(14,21,24,38)(15,26,28,42)(17,30,18,29)(20,33,35,43)(22,37,23,36)(25,39,34,44)(27,41,32,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,31,30)(20,34,35,25)(21,36,38,37)(26,40,42,41)(33,39,43,44);(6,9)(7,11)(13,17,19,18)(14,23,24,22)(15,27,28,32)(16,30,31,29)(20,25,35,34)(21,36,38,37)(26,41,42,40)(33,39,43,44)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513559);