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 | |
4355 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34,38,39)(35,36,40,43)(37,41,46,47)(42,44,45,48);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(34,39)(36,43)(41,47)(44,48);(2,7)(4,13)(6,8)(10,16)(19,27)(20,28)(22,30)(24,32)(33,35,38,40)(34,36,39,43)(37,45,46,42)(41,48,47,44);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,46)(47,48);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,36)(34,40)(35,39)(37,44)(38,43)(41,45)(42,47)(46,48);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,37,38,46)(34,41,39,47)(35,42,40,45)(36,44,43,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835888);