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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34)(35,40)(36,42)(37,41)(38,43)(39,46)(44,47)(45,48);(2,7)(3,10)(9,13)(11,16)(18,24)(19,20)(26,29)(27,28)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,47)(42,48);(2,7)(4,8)(9,13)(12,15)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36,38,37)(34,41,43,42)(35,44,46,45)(39,47,40,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37,38,36)(34,42,43,41)(35,45,46,44)(39,48,40,47)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026427);