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 | |
4172 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)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,6)(7,13)(8,11)(14,16)(18,25)(19,30)(20,27)(28,32)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,e^2=ghi,f^2=h,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, 9627262);