Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,33)(21,27)(22,26)(23,28)(25,36)(29,41)(30,44)(31,39)(32,40)(34,38)(35,43)(37,42)(45,47)(46,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,26,40)(20,34,36,29)(21,31,23,30)(25,41,33,38)(27,39,28,44)(35,46,37,45)(42,47,43,48);(10,14)(11,15)(17,20,22,36)(18,25,26,33)(19,29,32,34)(21,37,23,35)(24,38,40,41)(27,43,28,42)(30,45,31,46)(39,47,44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,26)(19,30)(21,23)(24,39)(25,33)(29,45)(31,32)(34,46)(35,37)(38,47)(40,44)(41,48);(2,7)(6,8)(17,21,22,23)(18,27,26,28)(19,31,32,30)(20,35,36,37)(24,39,40,44)(25,42,33,43)(29,45,34,46)(38,48,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7432464);