Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45257 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,25)(20,21)(22,26)(23,24)(27,32)(28,29)(30,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,2)(3,8)(4,5)(6,7)(9,14)(10,16)(11,13)(12,15)(18,24)(21,25)(26,31)(29,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(5,9)(8,15)(11,16)(17,19,23,20)(18,25,24,21)(22,28,30,27)(26,29,31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,20,23,19)(18,25,24,21)(22,28,30,27)(26,32,31,29)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,9)(3,10)(5,7)(6,14)(8,11)(12,13)(15,16)(17,21,23,25)(18,20,24,19)(22,29,30,32)(26,28,31,27)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,5,6,9)(2,4,7,14)(3,11,12,16)(8,13,15,10)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8278761);