Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021351 |
| 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 | #28174 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1623 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)(17,18,24,29,23,30,22,28)(19,31,34,25,35,45,33,42)(20,36,21,27,38,43,37,26)(32,44,46,41,47,40,39,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,32)(21,39)(22,34)(23,35)(24,33)(26,40)(27,44)(28,45)(29,31)(30,42)(36,41)(37,46)(38,47)(43,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,22,37,23,38,24,21)(18,26,29,36,30,27,28,43)(19,32,34,46,35,47,33,39)(25,40,31,41,42,44,45,48);(3,10)(6,9)(11,16)(14,15)(17,21,24,38,23,37,22,20)(18,27,29,43,30,26,28,36)(19,32,34,46,35,47,33,39)(25,41,45,40,42,48,31,44);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093914);