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 | |
4164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34)(35,39)(36,40)(37,45)(38,42)(41,47)(43,44)(46,48);(2,4)(7,11)(8,12)(14,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(33,35)(34,39)(36,42)(37,43)(38,40)(41,48)(44,45)(46,47);(3,9)(5,13)(7,11)(8,12)(19,27)(20,28)(22,30)(24,32)(33,35,38,40)(34,36,42,39)(37,46,47,43)(41,48,45,44);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835880);