Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357459 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60946 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,10)(3,5,9,6)(4,12,14,11)(8,13,16,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,41)(30,38)(32,42)(33,39)(34,40)(35,46)(37,45)(43,47)(44,48);(1,3,6,2,7,9,5,10)(4,11,15,8,14,12,13,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,42)(20,29,35,45,34,44,33,43)(25,37,39,41,40,47,46,48);(1,4,5,13,7,14,6,15)(2,8,3,11,10,16,9,12)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,41)(30,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,5,7,6)(2,9,10,3)(4,13,14,15)(8,12,16,11)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,42)(25,39,40,46)(29,43,44,45)(37,41,47,48);(4,14)(8,16)(11,12)(13,15)(20,34)(25,40)(29,44)(33,35)(37,47)(39,46)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264671);