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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(4,9)(8,14)(10,15)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,2,6,7)(3,8,12,14)(4,5,9,13)(10,11,15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(34,38)(35,42)(40,44)(43,48);(1,3,6,12)(2,8,7,14)(4,10,9,15)(5,11,13,16)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,5)(3,10)(6,9)(7,13)(8,11)(12,15)(14,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3)(2,8)(4,10)(5,11)(6,12)(7,14)(9,15)(13,16)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,5)(2,9)(3,11)(4,7)(6,13)(8,15)(10,14)(12,16)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442587);