Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1525 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)(10,14)(11,12)(17,18)(19,23)(20,24)(21,26)(22,25)(27,35)(28,42)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(43,47)(44,46)(45,48);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,30)(18,23,25,37)(20,31,32,27)(21,28,34,29)(24,38,39,35)(26,42,41,36)(33,44,45,43)(40,46,48,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,32)(18,24,25,39)(19,27,30,31)(21,33,34,45)(23,35,37,38)(26,40,41,48)(28,43,29,44)(36,46,42,47);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,16,14,15)(18,25)(19,28)(21,34)(23,36)(24,39)(27,43)(29,30)(31,44)(33,45)(35,46)(37,42)(38,47);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(20,32)(24,39)(27,31)(33,45)(35,38)(40,48)(43,44)(46,47)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,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, 7482873);