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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1666 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)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18,21,25,22,27,29,26)(19,28,31,23,32,36,42,37)(20,24,34,39,33,38,43,40)(30,41,45,35,44,47,48,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,18)(19,23)(20,24)(21,26)(22,27)(25,29)(28,31)(30,35)(32,37)(33,38)(34,40)(36,42)(39,43)(41,45)(44,46)(47,48);(2,7)(3,9)(8,11)(10,15)(17,19,29,42,22,32,21,31)(18,23,25,36,27,37,26,28)(20,30,43,48,33,44,34,45)(24,35,39,47,38,46,40,41);(3,9)(6,13)(10,15)(14,16)(17,20)(18,24)(19,30)(21,34)(22,33)(23,35)(25,39)(26,40)(27,38)(28,41)(29,43)(31,45)(32,44)(36,47)(37,46)(42,48);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(20,33)(24,38)(30,44)(34,43)(35,46)(39,40)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,e^2=h,f^2=i,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, 6405880);