Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,5,8,4)(9,10)(11,14)(12,16)(13,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,36)(29,40)(31,42)(32,37)(33,38)(34,41)(35,39)(43,46)(44,48)(45,47);(1,3,6,8)(2,4,7,5)(9,11,13,12)(10,14,15,16)(17,19,21,30)(18,23,24,27)(20,32,33,29)(22,28,35,31)(25,40,38,37)(26,42,39,36)(34,44,45,43)(41,46,47,48);(1,4,6,5)(2,3,7,8)(9,12,13,11)(10,14,15,16)(18,24)(19,28)(22,35)(23,36)(25,38)(27,42)(29,43)(30,31)(32,44)(34,45)(37,46)(40,48);(1,5,6,4)(2,3,7,8)(10,15)(14,16)(17,20,21,33)(18,25,24,38)(19,29,30,32)(22,34,35,45)(23,37,27,40)(26,41,39,47)(28,43,31,44)(36,46,42,48);(20,33)(25,38)(29,32)(34,45)(37,40)(41,47)(43,44)(46,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,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, 7416475);