Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,26)(21,27)(22,37)(23,25)(24,31)(29,45)(30,38)(32,44)(33,48)(34,42)(35,40)(36,43)(39,46)(41,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,32,21,30)(18,24,27,28,25,38,37,44)(20,33,34,47,35,29,36,46)(26,41,43,48,40,39,42,45);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,25)(24,38)(26,40)(27,37)(28,44)(39,41)(42,43)(45,48);(11,15)(13,16)(17,20)(18,26)(19,29)(21,36)(22,34)(23,35)(24,39)(25,40)(27,43)(28,45)(30,47)(31,46)(32,33)(37,42)(38,41)(44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,27,25,37)(19,30,32,31)(20,34,35,36)(24,28,38,44)(26,42,40,43)(29,46,33,47)(39,48,41,45)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384218);