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 | #26996 | 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)(17,18,22,29,23,27,31,28)(19,30,33,24,34,43,48,42)(20,35,37,41,38,44,36,25)(21,26,40,46,32,45,39,47);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,31,48,23,34,22,33)(18,24,29,43,27,42,28,30)(20,32,37,39,38,21,36,40)(25,26,44,47,41,45,35,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,25)(19,32)(21,34)(22,36)(23,38)(24,26)(27,41)(28,35)(29,44)(30,46)(31,37)(33,40)(39,48)(42,45)(43,47);(3,10)(6,9)(11,16)(14,15)(17,21,23,32)(18,26,27,45)(19,20,34,38)(22,40,31,39)(24,41,42,25)(28,47,29,46)(30,35,43,44)(33,37,48,36);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,27)(24,42)(25,41)(26,45)(28,29)(30,43)(35,44)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,d^2=i,e^2=h,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, 7090015);