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 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,13,15,9)(5,10,16,14)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,29)(30,32)(33,34,39,42)(35,40,43,38)(36,37,41,47)(44,45,48,46);(2,7)(3,12)(9,13)(10,14)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,39,43)(34,40,42,38)(36,44,41,48)(37,45,47,46);(2,7)(5,16)(8,11)(9,13)(21,29)(25,28)(26,32)(30,31)(35,43)(38,40)(44,48)(45,46);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,15)(13,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,8,16,11)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37)(34,41)(35,45)(36,42)(38,44)(39,47)(40,48)(43,46);(1,5,6,16)(2,10,7,14)(3,9,12,13)(4,8,15,11)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38)(34,35)(36,46)(37,44)(39,40)(41,45)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8192721);