Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18,21,26)(19,28,31,23)(20,33,34,25)(22,37,36,27)(24,38,43,35)(29,44,30,40)(32,41,48,45)(39,46,42,47);(1,2)(3,7)(4,8)(5,6)(10,13)(11,15)(17,19)(18,23)(20,30)(21,31)(22,32)(24,42)(25,40)(26,28)(27,41)(29,34)(33,44)(35,47)(36,48)(37,45)(38,46)(39,43);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,29)(21,34)(22,38)(23,39)(25,27)(26,43)(28,42)(30,31)(32,47)(33,37)(35,36)(40,45)(41,44)(46,48);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,40)(24,27)(26,33)(28,44)(29,31)(32,47)(36,38)(37,43)(39,45)(41,42)(46,48);(2,6)(7,8)(19,31)(20,34)(23,28)(24,43)(25,33)(32,48)(35,38)(41,45)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6680052);