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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,42)(36,45)(37,40)(38,46)(39,41)(43,44)(47,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,39,37)(34,40,41,42)(36,43,46,47)(38,44,45,48);(2,7)(3,9)(8,14)(10,15)(18,25)(19,20)(26,32)(27,28)(34,41)(36,46)(40,42)(43,47);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36)(34,38)(35,43)(37,47)(39,46)(40,44)(41,45)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,37,39,35)(34,40,41,42)(36,47,46,43)(38,44,45,48);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9720052);