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 | |
4243 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)(3,7)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(18,23)(19,25)(21,29)(28,32)(34,38)(35,42)(36,44)(46,47);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34)(35,39)(36,41)(37,38)(40,42)(43,46)(44,45)(47,48);(2,8)(3,9)(4,13)(11,16)(19,25)(20,27)(26,31)(28,32)(33,35,37,42)(34,39,38,40)(36,43,44,48)(41,46,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,20)(21,26)(22,25)(23,27)(24,28)(29,31)(30,32)(33,35)(34,40)(36,43)(37,42)(38,39)(41,47)(44,48)(45,46);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,34)(35,40)(36,41)(37,38)(39,42)(43,47)(44,45)(46,48);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,36,37,44)(34,41,38,45)(35,43,42,48)(39,46,40,47)Polycyclic presentation:
2^9|b^2=i,c^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034430);