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 | |
4248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(18,20)(19,25)(24,28)(26,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(34,37)(36,45)(39,41)(42,46);(2,6)(7,10)(8,12)(14,16)(19,25)(23,27)(26,31)(30,32)(33,35,38,43)(34,39,37,41)(36,42,45,46)(40,47,44,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35)(34,39)(36,46)(37,41)(38,43)(40,48)(42,45)(44,47);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,48)(43,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,48,46,47)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8762623);