Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18)(19,29)(20,34,38,44)(21,27)(22,26)(23,28)(24,31)(25,37,42,36)(30,39,47,48)(32,40)(33,43)(35,45,46,41);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,43)(20,35,36,30,38,46,37,47)(25,41,34,39,42,45,44,48);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(17,20,23,38)(18,25,28,42)(19,30,33,47)(21,36,22,37)(24,39,40,48)(26,34,27,44)(29,45,43,41)(31,46,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=eh,f^c=fh,g^c=ghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25630);