Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | #60924 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1591 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)(17,18,21,26,22,27,29,25)(19,28,33,23,32,40,44,30)(20,24,37,42,38,35,36,43)(31,45,48,39,47,41,34,46);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,29,44,22,32,21,33)(18,23,26,40,27,30,25,28)(20,34,37,31,38,48,36,47)(24,41,43,39,35,45,42,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,18)(19,30)(20,35)(21,25)(22,27)(23,32)(24,38)(26,29)(28,44)(31,39)(33,40)(34,41)(36,43)(37,42)(45,48)(46,47);(3,10)(6,9)(11,16)(14,15)(17,20)(18,24)(19,31)(21,37)(22,38)(23,39)(25,43)(26,42)(27,35)(28,45)(29,36)(30,46)(32,47)(33,48)(34,44)(40,41);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(19,32)(23,30)(28,40)(31,47)(33,44)(34,48)(39,46)(41,45)Polycyclic presentation:
2^9|a^2=g,b^2=fh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090006);