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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 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)(12,13)(14,16)(17,18,22,27)(19,29,33,24)(20,34,37,25)(21,26,23,28)(30,40,35,43)(31,41,48,47)(32,46,39,42)(36,45,38,44);(2,6)(7,8)(10,13)(11,16)(17,19,22,33)(18,24,27,29)(20,35,37,30)(21,32,23,39)(25,43,34,40)(26,42,28,46)(31,38,48,36)(41,45,47,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,37)(18,25,27,34)(19,30,33,35)(21,38,23,36)(24,40,29,43)(26,45,28,44)(31,39,48,32)(41,46,47,42);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,41)(26,44)(27,34)(28,45)(29,47)(30,32)(33,48)(35,39)(40,42)(43,46);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,21,22,23)(18,26,27,28)(19,32,33,39)(20,36,37,38)(24,42,29,46)(25,44,34,45)(30,48,35,31)(40,47,43,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7494697);