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 | |
4025 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,22,26)(19,25,29,23)(20,21,30,24)(27,31,32,28)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,12)(2,8,9,16)(4,13,15,6)(5,11,10,14)(17,19,22,29)(18,23,26,25)(20,27,30,32)(21,28,24,31)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,9)(3,12)(4,15)(11,14)(17,20,22,30)(18,24,26,21)(19,27,29,32)(23,31,25,28)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4)(2,10)(3,13)(5,9)(6,12)(7,15)(8,14)(11,16)(17,20)(18,21)(19,27)(22,30)(23,28)(24,26)(25,31)(29,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,10)(2,4,9,15)(3,14,12,11)(6,16,13,8)(17,19)(18,25)(20,27)(21,31)(22,29)(23,26)(24,28)(30,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6,7,13)(2,11,9,14)(3,4,12,15)(5,8,10,16)(17,21)(18,20)(19,28)(22,24)(23,27)(25,32)(26,30)(29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=gh,f^2=gh,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, 9619632);