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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3850 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,39,42)(35,43,46,40)(36,47,45,37)(38,44,48,41);(2,3)(7,9)(8,10)(14,15)(17,18,22,24)(19,20,27,25)(21,23,30,29)(26,28,32,31)(33,35,39,46)(34,40,42,43)(36,38,45,48)(37,44,47,41);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,34,39,42)(35,40,46,43)(36,47,45,37)(38,41,48,44);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,25)(18,19,24,27)(21,28,30,31)(23,26,29,32)(33,36,39,45)(34,37,42,47)(35,38,46,48)(40,44,43,41);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,37)(34,36)(35,44)(38,40)(39,47)(41,46)(42,45)(43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,38)(34,41)(35,45)(36,46)(37,40)(39,48)(42,44)(43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066128);