Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 |
| prime radical | #26993 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,24,29,23,28,22,30)(19,31,42,21,27,41,33,25)(20,34,37,26,39,44,38,43)(32,40,46,45,35,36,47,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,42,23,27,22,33)(18,25,30,41,28,21,29,31)(20,35,38,46,39,32,37,47)(26,40,44,45,43,36,34,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,40)(22,37)(23,39)(24,38)(25,36)(27,35)(28,43)(29,34)(30,44)(31,45)(33,46)(41,48)(42,47);(3,10)(6,9)(11,16)(14,15)(17,21)(18,27)(19,28)(20,36)(22,31)(23,25)(24,41)(26,32)(29,33)(30,42)(34,47)(35,43)(37,48)(38,45)(39,40)(44,46);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,28)(19,27)(26,43)(29,30)(32,35)(33,42)(34,44)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089618);