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 | |
3982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,18,23,27)(19,26,31,24)(20,22,21,25)(28,32,29,30)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(34,39)(35,42)(41,44)(43,48);(2,4)(5,13)(7,9)(10,16)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,32,26,30)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4,6,2)(3,7,11,9)(5,12,13,8)(10,14,16,15)(17,19)(18,26)(20,29)(21,28)(22,30)(23,31)(24,27)(25,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,28,26,29)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=g,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946332);