Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| 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 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2896 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)(19,27,30,23)(20,31,34,24)(21,35)(22,26)(25,36)(28,39)(29,38,43,41)(32,37)(33,40,46,44)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,43)(24,39,31,37)(25,38,35,41)(33,45,46,42)(40,48,44,47);(2,6)(7,13)(8,14)(15,16)(17,20)(18,24)(19,28)(21,33)(22,34)(23,37)(25,40)(26,31)(27,39)(29,42)(30,32)(35,44)(36,46)(38,47)(41,48)(43,45);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,21,22,36)(18,25,26,35)(19,29,30,43)(20,33,34,46)(23,38,27,41)(24,40,31,44)(28,42,32,45)(37,47,39,48);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(21,36)(25,35)(29,43)(33,46)(38,41)(40,44)(42,45)(47,48)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451250);