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 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18)(19,21)(20,25)(22,23)(24,26)(27,30)(28,31)(29,32)(33,34,38,40)(35,39,43,47)(36,41,45,48)(37,42,44,46);(2,6)(3,9)(8,13)(11,15)(18,23)(19,26)(25,31)(27,29)(33,35)(34,39)(36,37)(38,43)(40,47)(41,42)(44,45)(46,48);(3,9)(4,12)(7,10)(8,13)(18,23)(20,28)(21,24)(27,29)(34,40)(35,43)(37,44)(41,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,36,38,45)(34,41,40,48)(35,44,43,37)(39,46,47,42);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,25)(19,27)(21,30)(22,28)(23,31)(24,32)(26,29)(33,37,38,44)(34,42,40,46)(35,36,43,45)(39,41,47,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=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, 8947922);