Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,28,23,27,21,26)(19,29,31,38,32,42,36,24)(20,25,33,39,34,40,35,41)(30,43,45,48,46,47,44,37);(9,10)(11,16)(12,14)(13,15)(17,19,23,32)(18,24,27,38)(20,30,34,46)(21,31,22,36)(25,37,40,48)(26,29,28,42)(33,44,35,45)(39,47,41,43);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,41)(27,40)(28,39)(29,43)(31,45)(32,46)(36,44)(38,48)(42,47);(10,14)(13,16)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,33,34,35)(24,29,38,42)(25,39,40,41)(30,44,46,45)(37,47,48,43);(1,4,5,3)(2,6,7,8)(18,27)(24,38)(25,40)(26,28)(29,42)(37,48)(39,41)(43,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376334);