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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18,23,28)(19,29,20,25)(21,26,36,44)(22,37,24,27)(30,46,32,39)(31,40,33,41)(34,48,47,42)(35,45,38,43);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,25,28,29)(21,34,36,47)(22,31,24,33)(26,42,44,48)(27,40,37,41)(30,38,32,35)(39,43,46,45);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,19)(18,25,28,29)(21,32,36,30)(22,31,24,33)(26,39,44,46)(27,41,37,40)(34,35,47,38)(42,43,48,45);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,36)(24,38)(25,39)(27,43)(28,44)(29,46)(31,47)(33,34)(37,45)(40,42)(41,48);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,37)(19,31,20,33)(21,35,36,38)(25,40,29,41)(26,43,44,45)(30,47,32,34)(39,42,46,48)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,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, 7438097);