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 | |
4321 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)(3,6,9,14)(4,8,11,15)(10,13,12,16)(17,18)(19,27)(20,31)(21,26)(22,23)(24,30)(25,29)(28,32)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,5,9)(2,6,7,14)(4,10,11,12)(8,13,15,16)(17,19,23,30)(18,24,22,27)(20,28,29,21)(25,32,31,26)(36,44)(40,47)(42,48)(45,46);(2,7)(3,9)(4,11)(13,16)(18,22)(20,29)(21,28)(24,27)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4)(2,8)(3,10)(5,11)(6,13)(7,15)(9,12)(14,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,6)(4,12)(5,9)(7,14)(8,16)(10,11)(13,15)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,15)(12,16)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9401933);