Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1589 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)(9,10,12,14)(11,15,16,13)(17,18,22,28)(19,27,32,25)(20,34,36,45)(21,39,33,42)(23,30,24,29)(26,38,44,37)(31,47,48,43)(35,46,40,41);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,25,28,27)(20,35,36,40)(21,24,33,23)(26,43,44,47)(29,39,30,42)(31,38,48,37)(34,41,45,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,24,38,22,36,23,37)(18,26,30,45,28,44,29,34)(19,31,21,40,32,48,33,35)(25,41,39,47,27,46,42,43);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,24,32,22,33,23,19)(18,27,30,42,28,25,29,39)(20,35,38,31,36,40,37,48)(26,41,45,47,44,46,34,43);(2,6)(7,8)(10,14)(13,15)(18,28)(20,36)(25,27)(29,30)(31,48)(35,40)(37,38)(39,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fg,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499590);