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 |
| prime radical | #2668 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4032 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,27)(19,22,29,31)(20,26,30,24)(21,25,28,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,3)(7,9)(8,10)(14,15)(17,19,23,29)(18,22,27,31)(20,21,30,28)(24,32,26,25)(35,43)(36,46)(39,48)(40,45);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,23,30)(18,24,27,26)(19,28,29,21)(22,25,31,32)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21)(18,25)(19,20)(22,26)(23,28)(24,31)(27,32)(29,30);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22)(18,19)(20,25)(21,26)(23,31)(24,28)(27,29)(30,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,20)(18,26)(19,21)(22,25)(23,30)(24,27)(28,29)(31,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,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,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109164);