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 | |
4168 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(2,6)(7,9)(8,13)(14,15)(19,25)(22,29)(26,31)(30,32)(33,35,39,45)(34,40,37,47)(36,38,46,44)(41,43,42,48);(4,12)(8,13)(10,16)(14,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,39,46)(34,41,37,42)(35,44,45,38)(40,48,47,43);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,42)(35,38)(37,41)(39,46)(40,48)(43,47)(44,45);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37,39,34)(35,40,45,47)(36,41,46,42)(38,48,44,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,43)(35,36)(37,48)(39,44)(40,41)(42,47)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150938);