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 | |
1460 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)(19,24,20,25)(21,33,36,26)(22,27,38,46)(29,41,31,39)(30,40,32,42)(34,48,47,43)(35,45,37,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,28,25)(21,34,36,47)(22,32,38,30)(26,43,33,48)(27,42,46,40)(29,37,31,35)(39,45,41,44);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,19)(18,25,28,24)(21,31,36,29)(22,32,38,30)(26,41,33,39)(27,42,46,40)(34,35,47,37)(43,44,48,45);(10,14)(13,16)(17,21,23,36)(18,26,28,33)(19,29,20,31)(22,37,38,35)(24,39,25,41)(27,45,46,44)(30,47,32,34)(40,48,42,43);(1,4,5,3)(2,7,6,8)(19,20)(24,25)(29,31)(30,32)(34,47)(39,41)(40,42)(43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fi,d^2=i,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, 7435909);