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 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18)(19,28)(20,22)(21,31)(23,25)(24,32)(26,30)(27,29);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,19,23,32)(18,24,25,28)(20,29,30,21)(22,31,26,27)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(5,13)(8,15)(10,16)(12,14)(18,25)(22,26)(24,28)(27,31)(34,39)(37,45)(40,41)(43,48);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,26)(19,29)(21,32)(22,25)(23,30)(24,27)(28,31)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21,23,29)(18,27,25,31)(19,30,32,20)(22,24,26,28)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,22,23,26)(18,20,25,30)(19,31,32,27)(21,24,29,28)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300881);