Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18,23,28,22,26,21,27)(19,29,32,24,33,41,31,38)(20,25,34,40,36,39,35,42)(30,44,45,37,46,43,47,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,22,33)(18,24,26,38)(20,30,36,46)(21,31,23,32)(25,37,39,48)(27,29,28,41)(34,45,35,47)(40,43,42,44);(2,6)(7,13)(8,14)(15,16)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,37)(26,39)(27,40)(28,42)(29,43)(31,45)(32,47)(33,46)(38,48)(41,44);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(18,26)(24,38)(25,39)(27,28)(29,41)(37,48)(40,42)(43,44);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21,22,23)(18,27,26,28)(19,31,33,32)(20,34,36,35)(24,29,38,41)(25,40,39,42)(30,45,46,47)(37,43,48,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,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, 6451397);