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 | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(20,25)(21,26)(22,27)(23,28)(29,36)(30,38)(31,39)(32,40)(33,41)(34,42)(35,37)(43,48)(44,46)(45,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,36)(26,38)(27,37)(28,39)(32,44)(33,43)(34,45)(40,46)(41,48)(42,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,34)(18,25,28,42)(19,29,31,45)(21,32,22,33)(24,36,39,47)(26,40,27,41)(30,44,35,43)(37,48,38,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21)(18,26)(20,32)(22,23)(25,40)(27,28)(30,35)(33,34)(37,38)(41,42)(43,44)(46,48);(2,6)(7,8)(10,14)(13,15)(17,21,23,22)(18,27,28,26)(19,30,31,35)(20,33,34,32)(24,37,39,38)(25,40,42,41)(29,43,45,44)(36,46,47,48)Polycyclic presentation:
2^9|b^2=g,c^2=fgi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7476737);