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 | |
4256 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34)(35,41)(36,42)(37,47)(38,39)(40,45)(43,48)(44,46);(3,9)(7,13)(10,15)(14,16)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(34,39)(36,46)(40,41)(43,47);(2,6)(7,13)(8,11)(14,16)(18,25)(19,28)(20,30)(26,32)(33,35,38,45)(34,40,39,41)(36,43,46,47)(37,44,48,42);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,35)(34,41)(36,43)(37,44)(38,45)(39,40)(42,48)(46,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,36,38,46)(34,42,39,44)(35,43,45,47)(37,41,48,40);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,37,38,48)(34,43,39,47)(35,44,45,42)(36,40,46,41)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9634298);