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 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(11,16)(13,14)(17,18,23,28)(19,22,27,24)(20,21,26,25)(29,31,32,30)(33,34)(35,44)(36,41)(37,45)(38,46)(39,42)(40,43)(47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32)(33,35,40,42)(34,39,43,44)(36,45,47,38)(37,41,46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,20,23,26)(18,25,28,21)(19,29,27,32)(22,30,24,31)(33,36,40,47)(34,41,43,48)(35,45,42,38)(37,39,46,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,22)(20,25)(21,26)(23,28)(24,27)(29,30)(31,32)(33,37)(34,38)(35,41)(36,39)(40,46)(42,48)(43,45)(44,47);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,21,23,25)(18,26,28,20)(19,30,27,31)(22,29,24,32)(33,38)(34,37)(35,36)(39,41)(40,45)(42,47)(43,46)(44,48);(1,5,6,7)(2,9,10,3)(4,13,12,14)(8,16,15,11)(17,22)(18,27)(19,28)(20,30)(21,29)(23,24)(25,32)(26,31)(33,39)(34,42)(35,43)(36,46)(37,47)(38,41)(40,44)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366004);