Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,38)(20,31,32,24)(21,25,34,40)(27,35,29,37)(28,42,43,36)(33,45,46,39)(41,48,44,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,26,38)(20,28,32,43)(21,27,34,29)(24,36,31,42)(25,35,40,37)(33,41,46,44)(39,47,45,48);(10,14)(13,15)(19,27)(21,34)(23,35)(25,40)(28,41)(29,30)(33,46)(36,47)(37,38)(39,45)(42,48)(43,44);(2,5)(6,8)(11,16)(13,15)(17,20,22,32)(18,24,26,31)(19,28,30,43)(21,33,34,46)(23,36,38,42)(25,39,40,45)(27,41,29,44)(35,47,37,48);(1,3,4,7)(2,6,5,8)(20,32)(24,31)(28,43)(33,46)(36,42)(39,45)(41,44)(47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419454);