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 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,12,16)(11,15,14,13)(17,18)(19,20)(21,29)(22,24)(23,25)(26,30)(27,31)(28,32)(33,34)(35,37)(36,41)(38,48)(39,42)(40,46)(43,45)(44,47);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35,39,38)(34,37,42,48)(36,44,46,43)(40,45,41,47);(3,5)(4,12)(6,8)(9,16)(18,24)(20,23)(21,30)(27,32)(33,36,39,46)(34,40,42,41)(35,43,38,44)(37,47,48,45);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,36)(34,41)(35,44)(37,47)(38,43)(39,46)(40,42)(45,48);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,30)(31,32)(33,37)(34,35)(36,45)(38,42)(39,48)(40,44)(41,43)(46,47);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,38,39,35)(34,37,42,48)(36,43,46,44)(40,45,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8915035);