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 | |
4178 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)(17,18,23,21)(19,24,27,30)(20,25,28,29)(22,31,32,26)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,27)(18,24,21,30)(20,22,28,32)(25,31,29,26)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,6)(7,13)(8,11)(14,16)(18,21)(19,27)(20,28)(26,31)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,22)(21,29)(23,28)(24,31)(26,30)(27,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,24,27,30)(20,29,28,25)(22,31,32,26)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,26)(19,20)(21,31)(23,32)(24,29)(25,30)(27,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^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, 9150792);