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 | |
4142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,25)(19,24,29,32)(20,26,28,22)(21,31,30,27)(33,34)(35,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19)(18,24)(20,30)(21,28)(22,31)(23,29)(25,32)(26,27)(33,34,39,36)(35,40,37,43)(38,44,46,41)(42,48,45,47);(2,6)(7,10)(8,12)(14,16)(18,25)(19,29)(20,28)(27,31)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,28)(18,26,25,22)(19,30,29,21)(24,27,32,31)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,23,30)(18,27,25,31)(19,28,29,20)(22,24,26,32)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,26)(18,28,25,20)(19,27,29,31)(21,24,30,32)(33,38,39,46)(34,41,36,44)(35,42,37,45)(40,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663163);