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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,11)(4,10,15,14)(6,13,16,12)(17,18,22,24)(19,25,26,20)(21,23,29,30)(27,31,32,28);(1,3)(2,8)(4,13)(5,7)(6,14)(9,11)(10,16)(12,15)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,34,38,39)(35,40,43,48)(36,41,46,45)(37,42,47,44);(2,9)(3,5)(4,15)(6,16)(19,26)(20,25)(21,29)(23,30)(34,39)(35,43)(36,46)(37,47);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19,22,26)(18,20,24,25)(21,27,29,32)(23,28,30,31)(33,35)(34,40)(36,42)(37,45)(38,43)(39,48)(41,47)(44,46);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,36)(34,41)(35,44)(37,40)(38,46)(39,45)(42,43)(47,48);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21,22,29)(18,23,24,30)(19,27,26,32)(20,28,25,31)(33,37)(34,42)(35,45)(36,40)(38,47)(39,44)(41,43)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=gh,e^2=h,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, 8886969);