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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(18,23)(19,26)(21,30)(27,29)(33,34)(35,40)(36,38)(37,44)(39,42)(41,45)(43,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(35,42)(39,40)(43,48)(46,47);(2,6)(7,9)(8,13)(14,15)(19,26)(24,31)(27,29)(28,32)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,26)(18,24,23,31)(20,29,25,27)(21,28,30,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37,38,45)(34,41,36,44)(35,43,42,48)(39,46,40,47)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081812);