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 | |
4146 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(21,29)(25,30)(26,31)(27,32)(33,34)(35,41)(36,45)(37,42)(38,48)(39,40)(43,44)(46,47);(2,3)(4,11)(8,10)(12,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(34,40)(37,45)(41,48)(44,47);(2,3)(7,9)(8,10)(14,15)(18,23)(19,24)(25,30)(26,31)(33,35,39,38)(34,41,40,48)(36,46,42,43)(37,44,45,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36,39,42)(34,37,40,45)(35,43,38,46)(41,47,48,44);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37)(34,42)(35,44)(36,40)(38,47)(39,45)(41,43)(46,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,38,39,35)(34,41,40,48)(36,46,42,43)(37,47,45,44)Polycyclic presentation:
2^9|c^2=gh,d^2=gi,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^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019757);