Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2069 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(45,47)(46,48);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,32)(21,30)(22,35)(23,31)(25,40)(26,38)(27,39)(28,43)(29,33)(34,45)(36,46)(37,41)(42,47)(44,48);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,41)(27,42)(28,44)(30,45)(31,46)(32,35)(38,47)(39,48)(40,43);(2,3)(6,7)(10,14)(13,16)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,43)(25,41,42,44)(29,45,46,32)(37,47,48,40);(18,27)(19,31)(25,42)(26,28)(29,46)(30,35)(32,45)(41,44)Polycyclic presentation:
2^9|b^2=fg,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,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, 7411828);