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 | #42100 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,22,24)(19,27,28,23)(20,29,30,25)(21,26,31,32);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19,22,28)(18,23,24,27)(20,21,30,31)(25,26,29,32)(33,34,38,39)(35,40,43,48)(36,41,46,45)(37,42,47,44);(2,6)(3,4)(5,13)(15,16)(18,24)(20,30)(21,31)(23,27)(34,39)(35,43)(36,46)(37,47);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20,22,30)(18,25,24,29)(19,21,28,31)(23,26,27,32)(33,35)(34,40)(36,42)(37,45)(38,43)(39,48)(41,47)(44,46);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,21)(18,26)(19,20)(22,31)(23,25)(24,32)(27,29)(28,30)(33,36)(34,41)(35,44)(37,40)(38,46)(39,45)(42,43)(47,48);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,30)(31,32)(33,37)(34,42)(35,45)(36,40)(38,47)(39,44)(41,43)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8886787);