Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1678 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,23)(20,31)(21,25)(22,27)(24,33)(26,32)(28,41)(29,36)(30,37)(34,39)(35,44)(38,42)(40,45)(43,47)(46,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,37)(26,38)(27,36)(31,41)(32,42)(33,44)(34,43)(39,47)(40,46)(45,48);(2,6)(3,7)(10,13)(14,16)(17,20,32,45,22,34,21,33)(18,24,25,39,27,40,26,31)(19,28,42,48,29,43,30,44)(23,35,37,47,36,46,38,41);(9,11,12,15)(10,14,13,16)(19,29)(23,36)(28,43)(30,42)(35,46)(37,38)(41,47)(44,48);(3,7)(5,8)(17,21,22,32)(18,25,27,26)(19,30,29,42)(20,33,34,45)(23,37,36,38)(24,39,40,31)(28,44,43,48)(35,47,46,41)Polycyclic presentation:
2^9|c^2=fghi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670701);