Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432881 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2856 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)(9,12)(10,13)(11,14)(15,16)(18,23)(20,31)(21,33)(24,38)(25,40)(26,34)(27,36)(28,43)(29,35)(37,41)(39,47)(42,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,18)(19,27)(20,25)(21,34)(22,26)(23,33)(24,30)(28,42)(29,38)(31,41)(32,40)(35,36)(37,45)(39,43)(44,48)(46,47);(3,9)(6,12)(10,15)(13,16)(17,19,22,30)(18,24,26,27)(20,28,31,43)(21,35,33,29)(23,36,34,38)(25,39,41,42)(32,46,45,44)(37,47,40,48);(2,7)(6,12)(8,14)(13,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,37)(24,39)(26,41)(27,42)(29,44)(30,43)(33,45)(34,40)(35,46)(36,47)(38,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(20,31)(25,41)(28,43)(32,45)(37,40)(39,42)(44,46)(47,48)Polycyclic presentation:
2^9|c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343820);