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 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,8,13,15)(10,16,12,14)(17,18)(19,24)(20,26)(21,31)(22,27)(23,25)(28,30)(29,32);(1,3,5,11)(2,6,7,9)(4,10,13,12)(8,14,15,16)(17,19)(18,24)(20,25)(21,28)(22,29)(23,26)(27,32)(30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(2,7)(6,9)(8,15)(14,16)(18,25)(19,26)(21,27)(29,30)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,20,23,24)(18,26,25,19)(21,30,27,29)(22,32,31,28)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,27)(18,22,25,31)(19,28,26,32)(20,30,24,29)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,22)(18,27)(19,29)(20,28)(21,25)(23,31)(24,32)(26,30)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441636);