Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #23390 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,32)(25,35)(29,44)(30,43)(31,39)(34,41)(36,40)(38,47)(42,45)(46,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,39,37,43)(20,29,34,46,35,45,36,47)(25,38,41,42,33,48,40,44);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,41)(27,33)(28,42)(30,45)(31,47)(32,46)(37,40)(39,48)(43,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,39)(28,43)(29,45)(31,32)(38,48)(42,44)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,34,35,36)(24,28,39,43)(25,40,33,41)(29,46,45,47)(38,44,48,42)Polycyclic presentation:
2^9|b^2=f,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6443766);