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 | |
4219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,11,5)(4,7,12,15)(10,16,13,14)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(1,3,6,11)(2,5,8,9)(4,10,12,13)(7,14,15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(4,12)(7,15)(10,13)(14,16)(18,20)(23,25)(24,28)(30,31)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,3)(2,5)(4,13)(6,11)(7,16)(8,9)(10,12)(14,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,4)(2,7)(3,10)(5,14)(6,12)(8,15)(9,16)(11,13)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10314999);