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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,12)(3,13,15,9)(4,11,6,10)(5,14,16,7)(18,23)(20,24)(25,31)(28,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,8,15)(2,9,12,13)(4,14,6,7)(5,10,16,11)(18,23)(19,26)(25,31)(27,29)(35,42)(39,46)(43,47)(44,48);(1,4,8,6)(2,10,12,11)(3,14,15,7)(5,9,16,13)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,32,29,28)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,5,8,16)(2,7,12,14)(3,11,15,10)(4,9,6,13)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,30)(31,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6,8,4)(2,11,12,10)(3,7,15,14)(5,13,16,9)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29);(1,7)(2,5)(3,4)(6,15)(8,14)(9,10)(11,13)(12,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332703);