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 | #56044 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,24)(20,23)(21,25)(22,30)(26,32)(27,28)(29,31)(34,39)(35,43)(36,46)(42,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19)(18,24)(20,27)(21,22)(23,28)(25,30)(26,29)(31,32)(33,34,38,39)(35,44,43,40)(36,41,46,48)(37,47,45,42);(2,6)(4,12)(7,10)(11,15)(17,18,23,20)(19,27,28,24)(21,29,31,25)(22,30,32,26)(33,35,38,43)(34,40,39,44)(36,45,46,37)(41,47,48,42);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26)(33,36)(34,41)(35,45)(37,43)(38,46)(39,48)(40,47)(42,44);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21)(18,25)(19,22)(20,29)(23,31)(24,30)(26,27)(28,32)(33,37,38,45)(34,42,39,47)(35,36,43,46)(40,41,44,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,26)(19,21)(20,30)(23,32)(24,29)(25,27)(28,31)(33,36)(34,41)(35,45)(37,43)(38,46)(39,48)(40,47)(42,44)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=gi,e^2=g,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, 8946441);