Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #63034 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,5)(3,11,14,16)(4,10,6,9)(8,15,13,12)(17,18,23,28)(19,24,31,39)(20,32,34,42)(21,27,22,26)(25,36,40,33)(29,43,45,47)(30,41,35,38)(37,44,48,46);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,11)(10,16)(17,19,23,31)(18,24,28,39)(20,29,34,45)(21,30,22,35)(25,37,40,48)(26,38,27,41)(32,47,42,43)(33,44,36,46);(1,4,7,6)(2,9,5,10)(3,12,14,15)(8,11,13,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,32)(27,42)(28,40)(30,44)(31,45)(35,46)(38,47)(39,48)(41,43);(1,5,7,2)(3,13,14,8)(4,10,6,9)(11,12,16,15)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,45,46)(37,47,48,43);(1,4,7,6)(2,10,5,9)(3,12,14,15)(8,16,13,11)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257436);