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 | |
4306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,20)(21,29)(22,24)(23,27)(25,30)(26,31)(28,32)(33,34)(35,37)(36,38)(39,41)(40,44)(42,46)(43,47)(45,48);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,29)(30,32)(33,35,39,44)(34,40,41,37)(36,42,45,47)(38,43,48,46);(2,6)(3,9)(4,12)(14,16)(18,24)(20,23)(21,30)(26,32)(33,36,39,45)(34,38,41,48)(35,42,44,47)(37,46,40,43);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,18)(19,23)(20,27)(21,25)(22,24)(26,31)(28,32)(29,30)(33,36)(34,38)(35,42)(37,46)(39,45)(40,43)(41,48)(44,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,29,32)(33,37,39,40)(34,35,41,44)(36,43,45,46)(38,47,48,42);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,29)(27,32)(33,38)(34,36)(35,43)(37,47)(39,48)(40,42)(41,45)(44,46)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8957668);