Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,47)(31,43)(33,46)(34,42)(35,40)(36,44)(37,45)(38,41)(39,48);(1,3)(2,6)(4,7)(5,8)(17,19,22,32)(18,24,27,29)(20,31,21,35)(23,38,37,33)(25,40,26,43)(28,46,45,41)(30,36,48,34)(39,44,47,42);(10,13)(14,15)(17,20,22,21)(18,25,27,26)(19,30,32,48)(23,36,37,34)(24,39,29,47)(28,44,45,42)(31,33,35,38)(40,41,43,46);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,22,20)(18,26,27,25)(19,31,32,35)(23,36,37,34)(24,40,29,43)(28,44,45,42)(30,33,48,38)(39,41,47,46);(1,4,5,6)(2,3,7,8)(18,27)(20,21)(24,29)(28,45)(30,48)(31,35)(34,36)(41,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7418293);