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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1475 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)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,35)(29,45)(30,39)(31,40)(33,48)(34,44)(36,43)(37,42)(38,47)(41,46);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,36)(25,41)(26,39)(27,40)(28,43)(29,34)(32,48)(35,46)(37,47)(38,42)(44,45);(10,14)(13,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,42)(27,32)(28,44)(30,46)(31,47)(33,36)(39,48)(40,45)(41,43);(2,7)(3,8)(10,14)(13,16)(17,21,22,23)(18,26,27,28)(19,30,31,36)(20,34,35,37)(24,39,40,43)(25,42,32,44)(29,46,47,33)(38,48,45,41);(1,4,5,6)(2,3,7,8)(18,27)(24,40)(25,32)(26,28)(38,45)(39,43)(41,48)(42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414482);