Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500019 |
| 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 | C24 | C25 |
| prime radical | #62822 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2990 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,27,30,23)(20,31)(21,35)(22,26)(24,33)(25,37)(28,40,32,38)(29,39,36,42)(34,41)(43,47,45,48)(44,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,32)(21,36)(22,30)(24,40)(25,42)(26,27)(28,33)(29,37)(31,38)(34,43)(35,39)(41,47)(44,48)(45,46);(2,7)(3,9)(4,11)(12,16)(17,20)(18,24)(19,28)(21,34)(22,33)(23,38)(25,41)(26,31)(27,40)(29,43)(30,32)(35,44)(36,45)(37,46)(39,47)(42,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(20,33)(24,31)(28,32)(34,46)(38,40)(41,44)(43,45)(47,48);(3,9)(6,13)(10,15)(12,16)(17,21,22,37)(18,25,26,35)(19,29,30,36)(20,34,33,46)(23,39,27,42)(24,41,31,44)(28,43,32,45)(38,47,40,48)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432007);