Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1610 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,32,45)(20,25,22,37)(21,26,36,43)(24,29,39,31)(30,44,34,48)(33,40,35,42)(38,46,41,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,32,20,29)(18,24,25,28,27,39,37,45)(21,34,35,46,36,30,33,47)(26,41,40,48,43,38,42,44);(2,5)(6,8)(11,15)(13,16)(17,20,23,22)(18,25,27,37)(19,29,32,31)(21,35,36,33)(24,28,39,45)(26,42,43,40)(30,47,34,46)(38,48,41,44);(11,15)(13,16)(17,21)(18,26)(19,30)(20,33)(22,35)(23,36)(24,38)(25,40)(27,43)(28,44)(29,46)(31,47)(32,34)(37,42)(39,41)(45,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(21,36)(26,43)(30,34)(33,35)(38,41)(40,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383827);