Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #25614 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 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)(4,6)(5,8)(17,18)(19,29)(20,33)(21,27)(22,28)(23,38)(24,32)(25,34)(26,35)(30,40)(31,42)(36,41)(37,43)(39,46)(44,48)(45,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,39)(26,40)(27,42)(28,29)(33,45)(34,48)(35,47)(37,46)(38,41)(43,44);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,35)(18,25,28,43)(19,30,32,47)(21,34,23,37)(24,39,29,44)(26,38,33,27)(31,46,36,48)(40,41,45,42);(2,6)(5,7)(10,13)(14,16)(17,20)(18,26)(19,30)(21,37)(22,35)(23,34)(24,40)(25,27)(28,33)(29,45)(31,48)(32,47)(36,46)(38,43)(39,42)(41,44);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,28,38)(19,31,32,36)(20,34,35,37)(24,41,29,42)(25,26,43,33)(30,46,47,48)(39,45,44,40)Polycyclic presentation:
2^9|b^2=fi,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6871057);