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 | |
4211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(3,9)(8,13)(10,15)(17,18)(19,26)(20,24)(21,23)(22,29)(25,30)(27,31)(28,32)(33,34)(35,43)(36,39)(37,46)(38,47)(40,45)(41,48)(42,44);(1,2,6,5)(3,7,9,11)(4,8,12,13)(10,14,15,16)(19,20)(24,26)(27,28)(31,32)(33,35,39,45)(34,40,36,43)(37,44,48,38)(41,42,46,47);(4,12)(8,13)(10,15)(14,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,34,39,36)(35,40,45,43)(37,46,48,41)(38,42,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36,39,34)(35,40,45,43)(37,46,48,41)(38,47,44,42);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32)(33,37)(34,41)(35,44)(36,46)(38,45)(39,48)(40,42)(43,47);(1,5,6,2)(3,11,9,7)(4,13,12,8)(10,16,15,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,44)(34,42,36,47)(35,37,45,48)(40,46,43,41)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,e^2=i,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, 8026824);