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 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,6)(4,5,9,15)(12,13,16,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,39)(36,43)(41,45)(44,48);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(18,19)(23,25)(24,26)(30,31)(34,39)(35,40)(41,45)(42,47);(3,11)(6,8)(12,16)(13,14)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34,38,39)(35,36,40,43)(37,45,46,41)(42,48,47,44);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,48)(46,47);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=g,e^2=g,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, 8191701);