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 | |
2689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,28)(20,25)(21,27)(22,34)(23,26)(24,30)(29,44)(31,39)(32,41)(33,40)(35,38)(36,42)(37,46)(43,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,37)(26,38)(27,39)(28,34)(32,45)(33,47)(36,46)(40,48)(41,44)(42,43);(2,6)(7,13)(8,14)(15,16)(17,20,23,33)(18,25,26,40)(19,29,31,47)(21,32,22,36)(24,37,38,48)(27,41,34,42)(28,43,39,44)(30,45,35,46);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(18,26)(24,38)(25,40)(27,34)(28,39)(37,48)(41,42)(43,44);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21,23,22)(18,27,26,34)(19,30,31,35)(20,32,33,36)(24,39,38,28)(25,41,40,42)(29,45,47,46)(37,44,48,43)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,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, 6442244);