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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,23,28,38)(20,30,33,24)(21,25,36,44)(27,45,46,37)(29,39,34,42)(31,48,47,40)(32,43,35,41);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,31)(21,34)(22,28)(24,40)(25,42)(26,38)(27,32)(29,36)(30,48)(33,47)(35,46)(37,41)(39,44)(43,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,37)(25,39)(26,38)(30,45)(31,32)(33,46)(34,36)(35,47)(40,41)(42,44)(43,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,22,33)(18,24,26,30)(19,27,28,46)(21,35,36,32)(23,37,38,45)(25,43,44,41)(29,47,34,31)(39,48,42,40);(1,4,5,3)(2,7,8,6)(19,28)(23,38)(27,46)(29,34)(31,47)(37,45)(39,42)(40,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,d^2=fi,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, 7438777);