Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,27)(22,28)(23,29)(25,37)(30,44)(31,40)(32,41)(33,42)(34,43)(35,39)(36,46)(38,45)(47,48);(2,6)(7,8)(11,16)(13,15)(17,19,22,33)(18,24,28,42)(20,35,37,30)(21,34,23,32)(25,44,26,39)(27,43,29,41)(31,38,47,36)(40,45,48,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,38)(22,37)(23,36)(24,39)(26,28)(27,46)(29,45)(31,34)(32,47)(33,35)(40,41)(42,44)(43,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,26)(19,31)(21,36)(22,37)(23,38)(24,40)(25,28)(27,46)(29,45)(30,34)(32,35)(33,47)(39,41)(42,48)(43,44);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,27,28,29)(19,32,33,34)(20,36,37,38)(24,41,42,43)(25,45,26,46)(30,31,35,47)(39,48,44,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442710);