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 | |
4269 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,46,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,40)(34,37,36,42)(38,46,47,43)(41,48,44,45);(5,13)(8,12)(10,16)(14,15)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,34,39,36)(35,42,40,37)(38,41,47,44)(43,48,46,45);(1,4,6,2)(3,7,11,9)(5,8,13,12)(10,15,16,14)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36,39,34)(35,37,40,42)(38,44,47,41)(43,45,46,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37)(34,40)(35,36)(38,45)(39,42)(41,46)(43,44)(47,48);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275823);