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 | |
1413 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)(12,14)(13,16)(17,18)(19,28)(20,33)(21,36)(22,27)(23,26)(24,32)(25,35)(29,39)(30,40)(31,41)(34,43)(37,44)(38,42)(45,47)(46,48);(2,6)(7,8)(11,16)(13,15)(17,19,22,32)(18,24,27,28)(20,29,35,47)(21,31,23,38)(25,39,33,45)(26,41,36,42)(30,37,48,34)(40,43,46,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,39)(26,44)(27,33)(28,45)(30,38)(31,48)(32,47)(36,43)(40,41)(42,46);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,40)(26,43)(27,33)(28,46)(29,38)(31,47)(32,48)(36,44)(39,41)(42,45);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,36)(19,31,32,38)(20,34,35,37)(24,41,28,42)(25,43,33,44)(29,30,47,48)(39,46,45,40)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,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, 7442643);