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 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(33,34,38,42)(35,43,44,39)(36,40,46,47)(37,41,45,48);(2,6)(7,10)(8,12)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(35,44)(36,46)(39,43)(40,47);(2,6)(3,9)(4,13)(14,16)(18,22)(19,26)(20,29)(31,32)(33,35,38,44)(34,39,42,43)(36,37,46,45)(40,48,47,41);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36,38,46)(34,40,42,47)(35,45,44,37)(39,41,43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966561);