Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 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)(17,18,24,30)(19,31,33,28)(20,34,38,26)(21,29,23,27)(22,25,39,42)(32,45,35,47)(36,44,40,46)(37,43,48,41);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,39,24,33,23,22)(18,25,27,28,30,42,29,31)(20,35,40,37,38,32,36,48)(26,43,46,47,34,41,44,45);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,26)(19,32)(21,40)(22,37)(23,36)(24,38)(25,41)(27,46)(28,45)(29,44)(30,34)(31,47)(33,35)(39,48)(42,43);(2,6)(5,7)(10,14)(13,16)(17,21,24,23)(18,27,30,29)(19,22,33,39)(20,36,38,40)(25,31,42,28)(26,44,34,46)(32,48,35,37)(41,45,43,47);(2,6)(5,7)(17,22,23,33,24,39,21,19)(18,28,29,25,30,31,27,42)(20,37,36,35,38,48,40,32)(26,45,44,41,34,47,46,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507516);