Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2267 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)(16,20)(17,22)(18,32)(19,24)(21,27)(23,33)(26,39)(28,34)(29,35)(30,37)(31,43)(36,40)(38,42)(41,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,16,19,27,33,29)(14,20,23,25,24,35,32,21)(17,26,31,28,30,40,42,41)(22,34,38,39,37,44,43,36);(6,9)(7,11)(13,16,18,27,19,29,33,15)(14,21,23,20,24,25,32,35)(17,28,31,40,30,41,42,26)(22,36,38,34,37,39,43,44);(5,7,8,11)(6,10,9,12)(13,17,19,30)(14,22,24,37)(15,26,27,40)(16,28,29,41)(18,31,33,42)(20,34,35,44)(21,36,25,39)(23,38,32,43);(6,9)(10,12)(17,30)(22,37)(26,40)(28,41)(31,42)(34,44)(36,39)(38,43)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427663);