Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 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)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,40)(29,36)(30,35)(31,37)(32,38)(33,39)(41,48)(42,45)(43,46)(44,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,36)(26,40)(30,33)(31,43)(32,41)(35,39)(37,46)(38,48)(42,44)(45,47);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,31)(18,24,25,37)(19,27,29,43)(21,32,33,44)(23,34,36,46)(26,38,39,47)(28,41,30,42)(35,45,40,48);(9,11,12,16)(10,14,13,15)(18,25)(19,28)(21,33)(23,35)(24,37)(27,41)(29,30)(32,44)(34,45)(36,40)(42,43)(46,48);(1,4,5,3)(2,7,8,6)(19,29)(20,31)(23,36)(24,37)(28,30)(32,44)(35,40)(38,47)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7459795);