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 | |
3975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(19,25)(20,26)(21,31)(22,23)(24,29)(27,32)(28,30)(35,44)(37,47)(39,40)(42,48);(3,9)(7,13)(10,15)(14,16)(18,23)(20,30)(24,25)(27,31)(33,34,38,43)(35,40,44,39)(36,45,46,41)(37,42,47,48);(2,6)(7,13)(8,11)(14,16)(17,19,22,29)(18,24,23,25)(20,27,30,31)(21,28,32,26)(33,35,38,44)(34,39,43,40)(36,42,46,48)(37,41,47,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,24)(26,32)(30,31)(33,35)(34,40)(36,42)(37,41)(38,44)(39,43)(45,47)(46,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,30)(18,26,23,28)(19,27,29,31)(21,25,32,24)(33,36,38,46)(34,41,43,45)(35,42,44,48)(37,39,47,40);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,32)(18,27,23,31)(19,28,29,26)(20,24,30,25)(33,37)(34,42)(35,41)(36,40)(38,47)(39,46)(43,48)(44,45)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=h,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, 9634266);