Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,24)(21,26)(22,27)(23,29)(28,40)(30,43)(31,48)(32,41)(33,42)(34,44)(35,38)(36,45)(37,46)(39,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,23,26,25)(20,31,32,47)(22,35,34,30)(24,39,41,48)(27,43,44,38)(28,36,46,33)(37,45,40,42);(10,14)(13,15)(17,20,21,32)(18,24,26,41)(19,28,29,46)(22,36,34,33)(23,37,25,40)(27,45,44,42)(30,47,35,31)(38,48,43,39);(2,7)(3,8)(11,16)(13,15)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,40)(27,43)(31,33)(32,46)(34,35)(36,47)(37,41)(38,44)(39,45)(42,48);(1,4,5,6)(2,3,7,8)(19,29)(23,25)(28,46)(30,35)(31,47)(37,40)(38,43)(39,48)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7419052);