Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | #61345 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 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)(19,31,34,25,35,47,33,43)(20,36,38,26,39,48,37,44)(21,27,32,42,41,46,40,45);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,35)(18,25,30,43)(20,32,39,40)(21,37,41,38)(22,34,24,33)(26,42,44,45)(27,36,46,48)(28,47,29,31);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,33)(22,37)(23,39)(24,38)(25,42)(27,31)(28,36)(29,48)(30,44)(34,41)(35,40)(43,45)(46,47);(3,10)(6,9)(11,16)(14,15)(17,21,22,40,23,41,24,32)(18,27,28,45,30,46,29,42)(19,20,33,37,35,39,34,38)(25,26,31,36,43,44,47,48);(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=hi,d^2=f,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=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088156);