Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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×#3339 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2001 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)(7,11)(8,10)(9,12)(14,19)(20,35)(21,27)(22,33)(23,37)(36,44)(38,41)(39,40);(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,23,25,37)(16,27,30,21)(18,33,32,22)(20,34,35,26)(24,41,42,38)(28,44,43,36)(29,40,31,39);(2,4)(13,15,17,25)(14,20,19,35)(16,28,30,43)(18,34,32,26)(21,38,27,41)(22,37,33,23)(24,31,42,29)(36,39,44,40);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,24)(17,30)(18,29)(19,27)(20,36)(22,40)(23,38)(25,42)(26,28)(31,32)(33,39)(34,43)(35,44)(37,41);(5,8,9,7)(6,10,12,11)(14,22)(16,29)(18,32)(19,33)(20,37)(21,27)(23,35)(24,28)(26,34)(30,31)(38,41)(42,43)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485756);