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 | #38512 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,11,13,7)(4,14,12,8)(5,9,15,16)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,3,6,13)(2,7,10,11)(4,5,12,15)(8,9,14,16)(19,29)(20,31)(24,28)(25,30)(34,38)(39,41)(40,44)(46,47);(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,4)(2,8)(3,5)(6,12)(7,9)(10,14)(11,16)(13,15)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,13)(12,14)(15,16)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,5)(2,9)(3,12)(4,13)(6,15)(7,14)(8,11)(10,16)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8352904);