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 | |
4116 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)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,22,23)(19,25,30,24)(20,26,31,29)(21,28,32,27)(33,34,38,36)(35,39,43,40)(37,44,45,41)(42,47,48,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,23)(20,31)(24,25)(27,28)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(2,6)(7,9)(8,12)(14,15)(17,19,22,30)(18,24,23,25)(20,28,31,27)(21,29,32,26)(33,35,38,43)(34,40,36,39)(37,42,45,48)(41,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,24)(26,32)(27,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,26)(19,28)(21,24)(22,31)(23,29)(25,32)(27,30)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21,22,32)(18,27,23,28)(19,29,30,26)(20,25,31,24)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=h,c^2=gi,e^2=i,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, 9627686);