Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636852 |
| 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 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,33)(21,28)(22,26)(23,27)(24,31)(25,36)(30,44)(32,39)(34,40)(35,41)(37,42)(38,48)(43,45)(46,47);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,38)(26,29)(27,42)(28,39)(33,47)(34,46)(35,48)(36,45)(40,44)(41,43);(11,15)(13,16)(17,20,23,36,22,34,21,35)(18,25,28,33,26,41,27,40)(19,30,32,45,31,46,37,48)(24,38,39,47,29,43,42,44);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,26)(19,31)(20,34)(27,28)(32,37)(35,36)(38,43)(44,47);(2,5)(6,7)(10,14)(13,16)(17,21,22,23)(18,27,26,28)(19,32,31,37)(20,35,34,36)(24,39,29,42)(25,40,41,33)(30,45,46,48)(38,47,43,44)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6544604);