Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| 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 | #62864 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18,22,26)(19,23,29,36)(20,30)(21,25,33,38)(24,32)(27,34)(28,35,41,46)(31,43)(37,44)(39,45)(40,42)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,35)(26,36)(30,42)(31,39)(32,40)(33,41)(37,45)(38,46)(43,48)(44,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,27)(21,31)(22,32)(23,34)(25,37)(26,30)(28,39)(29,40)(33,44)(35,45)(36,42)(38,43)(41,47)(46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,33)(18,25,26,38)(19,28,29,41)(20,31,32,44)(23,35,36,46)(24,37,30,43)(27,39,40,47)(34,45,42,48);(2,8)(6,9)(7,16)(14,15)(21,33)(25,38)(28,41)(31,44)(35,46)(37,43)(39,47)(45,48)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=hi,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436528);