Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 | |
1515 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)(19,23,29,37)(20,30,31,24)(21,25,33,39)(27,35,34,40)(28,42,43,36)(32,44,45,38)(41,48,46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(19,27)(21,33)(23,35)(25,39)(28,41)(29,34)(32,45)(36,47)(37,40)(38,44)(42,48)(43,46);(10,14)(13,15)(17,19,22,29)(18,23,26,37)(20,28,31,43)(21,34,33,27)(24,36,30,42)(25,40,39,35)(32,46,45,41)(38,48,44,47);(2,5)(6,8)(11,16)(13,15)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,38)(26,30)(27,41)(29,43)(33,45)(34,46)(35,47)(37,42)(39,44)(40,48);(1,3,4,7)(2,6,5,8)(19,29)(20,31)(23,37)(24,30)(27,34)(32,45)(35,40)(38,44)Polycyclic presentation:
2^9|a^2=fi,c^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,e^c=ei,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419512);