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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(7,10)(8,12)(14,16)(17,18,22,26)(19,27,28,23)(20,24,31,29)(21,25,30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(3,9)(7,10)(11,15)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,31)(27,28)(30,32)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,19,22,28)(18,23,26,27)(20,30,31,21)(24,25,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,20,22,31)(18,24,26,29)(19,21,28,30)(23,32,27,25)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,18)(19,23)(20,24)(21,32)(22,26)(25,30)(27,28)(29,31)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21)(18,25)(19,20)(22,30)(23,29)(24,27)(26,32)(28,31)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646491);