Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473167 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1570 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)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(18,23)(19,27)(21,32)(24,36)(25,34)(28,40)(29,41)(31,43)(33,38)(37,46)(42,48)(44,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18,21,26,22,23,32,35)(19,24,27,38,29,33,41,36)(20,25,31,39,30,34,43,45)(28,37,40,47,42,44,48,46);(3,9)(6,13)(10,11)(14,15)(17,19,21,27,22,29,32,41)(18,24,26,38,23,33,35,36)(20,28,31,40,30,42,43,48)(25,37,39,47,34,44,45,46);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20)(18,25)(19,28)(21,31)(22,30)(23,34)(24,37)(26,39)(27,40)(29,42)(32,43)(33,44)(35,45)(36,46)(38,47)(41,48);(2,8)(6,13)(7,16)(14,15)(20,30)(25,34)(28,42)(31,43)(37,44)(39,45)(40,48)(46,47)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391805);