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 | #45643 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,28)(19,26,21,24)(20,25,31,29)(22,30,32,27)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(19,21)(22,32)(24,26)(27,30)(34,40)(35,44)(37,47)(42,46);(2,6)(7,10)(8,12)(14,16)(17,19,23,21)(18,24,28,26)(20,30,31,27)(22,29,32,25)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,25)(19,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,19)(18,26,28,24)(20,27,31,30)(22,25,32,29)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,22,23,32)(18,27,28,30)(19,29,21,25)(20,26,31,24)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8783365);