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 | |
4241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,14)(15,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,30)(26,31)(29,32)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,14)(17,19,22,27)(18,23,20,24)(21,26,30,29)(25,31,28,32)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(18,20)(23,24)(25,28)(31,32)(34,39)(35,42)(40,45)(43,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,3,5,11)(2,7,6,9)(4,12,13,10)(8,16,14,15)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,5,13)(2,8,6,14)(3,10,11,12)(7,15,9,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10109200);