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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,23,29)(19,25,32,41)(20,33,31,26)(21,35,37,27)(22,38,24,28)(30,46,48,40)(34,42,47,43)(36,44,39,45);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,24)(21,34)(22,31)(23,32)(26,38)(27,43)(28,33)(29,41)(30,36)(35,42)(37,47)(39,48)(40,45)(44,46);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,23,31)(18,26,29,33)(19,22,32,24)(21,34,37,47)(25,28,41,38)(27,42,35,43)(30,36,48,39)(40,44,46,45);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21)(18,27)(19,30)(20,34)(22,36)(23,37)(24,39)(25,40)(26,42)(28,44)(29,35)(31,47)(32,48)(33,43)(38,45)(41,46);(2,8)(6,7)(17,22,23,24)(18,28,29,38)(19,31,32,20)(21,36,37,39)(25,33,41,26)(27,44,35,45)(30,47,48,34)(40,43,46,42)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fi,d^2=f,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,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433560);