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 | #35944 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,7)(8,15)(9,11)(10,14)(12,16)(17,18)(19,23)(20,21)(22,25)(24,26)(27,28)(29,32)(30,31)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,46)(47,48);(2,7)(4,11)(8,13)(12,14)(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,30)(31,32)(33,35,39,42)(34,38,43,44)(36,45,41,48)(37,46,40,47);(2,7)(3,9)(8,13)(10,16)(19,26)(23,24)(27,29)(28,32)(33,36,39,41)(34,40,43,37)(35,45,42,48)(38,47,44,46);(1,3,6,9)(2,4,7,11)(5,10,15,16)(8,12,13,14)(17,19)(18,24)(20,29)(21,28)(22,26)(23,25)(27,30)(31,32)(33,36)(34,40)(35,45)(37,43)(38,47)(39,41)(42,48)(44,46);(1,4)(2,3)(5,14)(6,11)(7,9)(8,16)(10,13)(12,15)(17,20,22,30)(18,21,25,31)(19,27,26,29)(23,28,24,32)(33,37,39,40)(34,41,43,36)(35,46,42,47)(38,48,44,45);(1,5,6,15)(2,8,7,13)(3,10,9,16)(4,12,11,14)(17,21)(18,20)(19,28)(22,31)(23,27)(24,29)(25,30)(26,32)(33,38)(34,42)(35,43)(36,47)(37,45)(39,44)(40,48)(41,46)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9110878);