Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500226 |
| 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 |
| prime radical | #62878 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2868 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)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,35,22,24,21,27)(19,28,34,25,30,41,31,37)(20,26,36,46,32,38,33,40)(29,42,45,39,43,48,44,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,24)(21,23)(25,37)(26,38)(31,34)(33,36)(39,47)(44,45);(2,7)(3,9)(4,11)(12,16)(17,19)(18,25)(20,29)(21,34)(22,30)(23,31)(24,37)(26,39)(27,41)(28,35)(32,43)(33,45)(36,44)(38,47)(40,48)(42,46);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(17,20)(18,26)(19,29)(21,33)(22,32)(23,36)(24,38)(25,39)(27,40)(28,42)(30,43)(31,44)(34,45)(35,46)(37,47)(41,48);(3,9)(6,13)(10,15)(12,16)(20,32)(26,38)(29,43)(33,36)(39,47)(40,46)(42,48)(44,45)Polycyclic presentation:
2^9|a^2=ghi,d^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432088);