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 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,13)(8,11)(14,16)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(33,34,40,44)(35,43,38,39)(36,42,37,41)(45,46,47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(33,35,40,38)(34,39,44,43)(36,45,37,47)(41,46,42,48);(3,9)(7,13)(10,15)(14,16)(18,25)(19,21)(26,32)(27,29)(33,36,40,37)(34,41,44,42)(35,45,38,47)(39,46,43,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,37,40,36)(34,42,44,41)(35,45,38,47)(39,46,43,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,38,40,35)(34,43,44,39)(36,45,37,47)(41,46,42,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,39,40,43)(34,35,44,38)(36,46,37,48)(41,45,42,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441861);