Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,11)(4,8)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,28)(20,33)(21,38)(22,27)(23,26)(24,32)(25,37)(29,42)(30,41)(31,39)(34,45)(35,44)(36,43)(40,46)(47,48);(1,3,6,13)(2,7,9,11)(4,12,5,14)(8,15,10,16)(17,19,22,32)(18,24,27,28)(20,34,36,46)(21,30,23,39)(25,42,44,47)(26,41,38,31)(29,37,48,35)(33,40,43,45);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,36)(18,25,27,44)(19,29,32,48)(21,35,23,37)(24,40,28,45)(26,43,38,33)(30,46,39,34)(31,47,41,42);(2,9)(7,11)(8,10)(15,16)(17,21,22,23)(18,26,27,38)(19,30,32,39)(20,35,36,37)(24,41,28,31)(25,43,44,33)(29,46,48,34)(40,42,45,47);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,18)(19,31)(20,25)(21,26)(22,27)(23,38)(24,30)(28,39)(29,47)(32,41)(33,37)(34,45)(35,43)(36,44)(40,46)(42,48)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582077);