Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353473 |
| 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 | #63060 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,3)(4,12,14,16)(5,9,6,11)(8,15,10,13)(17,18)(19,31)(20,37)(21,27)(22,28)(23,29)(24,30)(25,35)(26,38)(32,42)(33,45)(34,44)(36,48)(39,46)(40,47)(41,43);(1,3,7,2)(4,10,14,8)(5,11,6,9)(12,15,16,13)(17,19)(18,25)(20,32)(21,33)(22,34)(23,35)(24,36)(26,42)(27,43)(28,31)(29,48)(30,44)(37,46)(38,40)(39,41)(45,47);(1,4,7,14)(2,8,3,10)(5,13,6,15)(9,16,11,12)(17,20,24,33)(18,26,30,43)(19,32,36,21)(22,38,23,41)(25,42,44,27)(28,45,29,37)(31,47,48,46)(34,40,35,39);(1,5,7,6)(2,9,3,11)(4,13,14,15)(8,16,10,12)(17,21,24,32)(18,27,30,42)(19,33,36,20)(22,40,23,39)(25,43,44,26)(28,47,29,46)(31,45,48,37)(34,38,35,41);(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,33,41)(21,39,32,40)(25,31,44,48)(26,45,43,37)(27,46,42,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260760);