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 | |
4256 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)(8,9)(10,14)(11,15)(12,16)(17,18,22,24)(19,23,26,20)(21,29,30,25)(27,28,32,31)(33,34,39,36)(35,40,42,37)(38,44,46,41)(43,45,48,47);(2,7)(3,9)(10,13)(11,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(2,7)(4,8)(10,13)(12,15)(18,24)(19,26)(25,29)(27,32)(35,42)(37,40)(43,48)(45,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032683);