Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632074 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1461 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)(17,18,22,25)(19,27,28,40)(20,24,31,39)(21,33,35,26)(23,34,36,30)(29,41,42,48)(32,45,44,38)(37,46,47,43);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(19,28)(23,36)(27,40)(29,42)(30,34)(37,47)(41,48)(43,46);(3,8)(5,7)(10,13)(14,16)(17,19,22,28)(18,23,25,36)(20,29,31,42)(21,34,35,30)(24,37,39,47)(26,40,33,27)(32,46,44,43)(38,48,45,41);(2,6)(5,7)(10,13)(14,16)(17,20)(18,24)(19,29)(21,32)(22,31)(23,37)(25,39)(26,38)(27,41)(28,42)(30,43)(33,45)(34,46)(35,44)(36,47)(40,48);(9,11,12,15)(10,14,13,16)(18,25)(20,31)(23,36)(26,33)(27,40)(29,42)(32,44)(43,46)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6560637);