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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 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)(11,14)(12,16)(13,15)(17,18,22,27)(19,28,32,24)(20,25,35,42)(21,36,23,26)(29,45,48,38)(30,46,47,39)(31,40,33,44)(34,43,37,41);(2,6)(7,8)(9,11,14,10)(12,16,13,15)(17,19,22,32)(18,24,27,28)(20,29,35,48)(21,33,23,31)(25,38,42,45)(26,44,36,40)(30,37,47,34)(39,41,46,43);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,38)(26,43)(27,42)(28,45)(30,31)(32,48)(33,47)(36,41)(39,44)(40,46);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,39)(26,41)(27,42)(28,46)(29,31)(32,47)(33,48)(36,43)(38,44)(40,45);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,11,15)(17,21,22,23)(18,26,27,36)(19,31,32,33)(20,34,35,37)(24,40,28,44)(25,41,42,43)(29,47,48,30)(38,39,45,46)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fgi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7450619);