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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(9,10,13,11)(12,15,16,14)(17,18,23,29)(19,30,34,25)(20,27,21,26)(22,38,24,28)(31,40,35,44)(32,41,48,47)(33,42,39,43)(36,45,37,46);(2,5)(6,7)(9,11,13,10)(12,14,16,15)(17,19,23,34)(18,25,29,30)(20,35,21,31)(22,33,24,39)(26,44,27,40)(28,42,38,43)(32,36,48,37)(41,46,47,45);(9,12,13,16)(10,14,11,15)(17,20,23,21)(18,26,29,27)(19,31,34,35)(22,37,24,36)(25,40,30,44)(28,46,38,45)(32,33,48,39)(41,43,47,42);(10,11)(14,15)(17,21,23,20)(18,27,29,26)(19,32,34,48)(22,37,24,36)(25,41,30,47)(28,46,38,45)(31,33,35,39)(40,43,44,42);(1,3,4,8)(2,6,5,7)(17,22,23,24)(18,28,29,38)(19,33,34,39)(20,36,21,37)(25,42,30,43)(26,45,27,46)(31,32,35,48)(40,47,44,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7452831);