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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,36)(22,27)(23,26)(25,35)(28,44)(29,48)(30,43)(31,42)(33,39)(34,45)(37,46)(38,41)(40,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,42)(20,33,35,28)(21,30,23,38)(25,44,32,39)(26,41,36,43)(29,37,47,34)(40,45,48,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,25)(19,28)(21,37)(22,35)(23,34)(24,39)(26,46)(27,32)(29,30)(31,33)(36,45)(38,47)(40,43)(41,48)(42,44);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,40)(26,45)(27,32)(28,30)(31,47)(33,38)(36,46)(39,43)(41,44)(42,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,36)(19,30,31,38)(20,34,35,37)(24,41,42,43)(25,45,32,46)(28,47,33,29)(39,40,44,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fgi,e^2=fg,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, 7496766);