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 | |
4278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(7,9)(12,16)(14,15)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,38,39)(35,40,43,37)(36,44,47,41)(42,46,45,48);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,19,22,27)(18,20,23,25)(21,30,31,26)(24,32,29,28)(34,39)(37,40)(41,44)(46,48);(2,3)(7,9)(8,10)(14,15)(18,23)(20,25)(24,29)(28,32)(33,35,38,43)(34,40,39,37)(36,42,47,45)(41,48,44,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20,22,25)(18,19,23,27)(21,28,31,32)(24,26,29,30)(33,35)(34,37)(36,45)(38,43)(39,40)(41,48)(42,47)(44,46);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,37)(34,35)(36,46)(38,40)(39,43)(41,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=hi,e^2=gh,f^2=h,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, 8750774);