Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632082 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,40)(29,43)(33,39)(36,37)(38,48)(41,42)(44,45)(46,47);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(18,23)(19,28)(20,32)(26,35)(30,36)(33,47)(38,45)(41,43);(3,8)(5,7)(10,13)(14,16)(17,19,21,28)(18,24,23,37)(20,29,32,42)(22,36,34,30)(25,38,31,45)(26,40,35,27)(33,48,47,44)(39,43,46,41);(2,6)(5,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,32)(22,33)(23,31)(24,38)(26,39)(27,41)(28,42)(30,44)(34,47)(35,46)(36,48)(37,45)(40,43);(9,11,12,15)(10,14,13,16)(18,23)(20,32)(24,37)(26,35)(27,40)(29,42)(33,47)(44,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6560679);