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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1405 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)(17,18,23,26)(19,27,30,45)(20,32,34,25)(21,37,31,24)(22,40,39,28)(29,43,33,46)(35,42,38,41)(36,44,47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,31,23,30,39,21)(18,24,28,27,26,37,40,45)(20,33,36,38,34,29,47,35)(25,42,44,43,32,41,48,46);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,29)(21,38)(22,36)(23,34)(24,41)(26,32)(27,46)(28,44)(30,33)(31,35)(37,42)(39,47)(40,48)(43,45);(2,6)(3,8)(10,14)(13,16)(18,26)(19,30)(20,34)(21,31)(28,40)(36,47)(41,42)(43,46);(2,6)(5,7)(17,21,39,30,23,31,22,19)(18,27,40,24,26,45,28,37)(20,35,47,29,34,38,36,33)(25,43,48,42,32,46,44,41)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,e^2=fg,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^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507845);