Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60956 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(17,18,21,26,22,25,32,38)(19,23,29,35,28,34,41,45)(20,24,30,37,31,36,42,46)(27,33,40,44,39,43,47,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,34)(26,35)(30,40)(31,39)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(1,3,5,10)(2,7,9,14)(4,11,12,16)(6,13,8,15)(17,20,21,30,22,31,32,42)(18,24,26,37,25,36,38,46)(19,27,29,40,28,39,41,47)(23,33,35,44,34,43,45,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48);(3,10)(7,14)(11,16)(13,15)(18,25)(23,34)(24,36)(26,38)(33,43)(35,45)(37,46)(44,48)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,d^b=di,f^b=fh,d^c=dh,e^c=eh,e^d=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252467);