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 | |
4225 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)(4,10)(5,15)(6,16)(7,9)(11,14)(12,13)(18,23)(19,25)(21,29)(28,32)(34,37)(35,41)(36,44)(47,48);(1,3)(2,8)(4,7)(5,11)(6,12)(9,10)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34,38,37)(35,39,41,43)(36,45,44,40)(42,48,46,47);(2,9)(3,4)(5,16)(12,14)(19,25)(20,27)(26,31)(28,32)(33,35,38,41)(34,39,37,43)(36,46,44,42)(40,48,45,47);(1,4,7,3)(2,10,9,8)(5,13,16,11)(6,14,15,12)(17,19)(18,20)(21,26)(22,25)(23,27)(24,28)(29,31)(30,32)(33,36,38,44)(34,40,37,45)(35,42,41,46)(39,47,43,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,37,38,34)(35,43,41,39)(36,45,44,40)(42,48,46,47)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=h,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, 8034552);