Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | #61423 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(17,18,22,27,23,28,21,26)(19,29,31,40,33,44,32,24)(20,34,37,42,38,25,36,43)(30,45,35,39,47,41,46,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,41)(26,29)(27,44)(28,40)(30,37)(34,45)(36,47)(38,46)(39,43)(42,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,31)(30,39)(32,44)(33,40)(34,38)(35,41)(36,42)(37,43)(45,46)(47,48);(3,10)(6,9)(11,16)(14,15)(17,20,22,37,23,38,21,36)(18,25,27,43,28,34,26,42)(19,30,31,35,33,47,32,46)(24,39,29,41,40,48,44,45);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,40,44)(25,42,34,43)(30,35,47,46)(39,41,48,45)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088173);