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 |
| prime radical | #57398 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3987 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)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(34,39)(37,45)(40,46)(44,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,19,22,20)(18,23,24,25)(21,28,30,27)(26,32,29,31)(36,43)(40,46)(42,47)(44,48);(2,6)(7,9)(8,12)(14,15)(18,24)(23,25)(26,29)(31,32)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,30)(18,26,24,29)(19,27,20,28)(23,31,25,32)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=ghi,e^2=h,f^2=g,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, 8662536);