Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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 | |
1576 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,32)(21,27)(22,24)(23,28)(26,33)(29,40)(30,44)(31,37)(34,42)(35,41)(36,38)(39,46)(43,47)(45,48);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,33)(21,23)(25,38)(30,45)(31,36)(37,40)(39,48)(41,42)(43,46)(44,47);(10,14)(13,15)(17,19,22,31)(18,25,24,37)(20,30,33,46)(21,29,23,36)(26,39,32,44)(27,40,28,38)(34,45,35,43)(41,47,42,48);(2,3)(6,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,32)(25,39)(27,42)(28,41)(29,43)(31,46)(36,45)(37,44)(38,47)(40,48);(1,4,5,8)(2,6,3,7)(17,21,22,23)(18,27,24,28)(19,29,31,36)(20,34,33,35)(25,40,37,38)(26,41,32,42)(30,45,46,43)(39,47,44,48)Polycyclic presentation:
2^9|b^2=fi,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419711);