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 | |
4091 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)(3,10)(4,14)(5,8)(6,15)(7,9)(11,16)(12,13)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,39,41)(35,37,43,36)(38,40,47,46)(42,45,48,44);(1,3,7,5)(2,8,9,10)(4,12,15,11)(6,13,14,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28)(33,35,39,43)(34,36,41,37)(38,42,47,48)(40,44,46,45);(1,4)(2,6)(3,11)(5,12)(7,15)(8,16)(9,14)(10,13)(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,36,39,37)(34,35,41,43)(38,44,47,45)(40,42,46,48);(1,4)(2,6)(3,12)(5,11)(7,15)(8,13)(9,14)(10,16)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,34)(35,36)(37,43)(38,46)(39,41)(40,47)(42,45)(44,48);(1,5,7,3)(2,8,9,10)(4,12,15,11)(6,16,14,13)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,37,39,36)(34,35,41,43)(38,45,47,44)(40,42,46,48);(1,6)(2,4)(3,13)(5,16)(7,14)(8,12)(9,15)(10,11)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,38)(34,40)(35,42)(36,44)(37,45)(39,47)(41,46)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319315);