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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,12,6,8)(4,14,15,9)(5,10,16,13)(18,20)(23,26)(24,28)(30,32);(1,3,7,6)(2,8,11,12)(4,10,15,13)(5,9,16,14)(19,25)(23,26)(27,31)(30,32)(34,39)(35,41)(40,47)(42,48);(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,38,39)(35,37,41,46)(36,40,44,47)(42,45,48,43);(1,4,7,15)(2,9,11,14)(3,13,6,10)(5,12,16,8)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,35)(34,37)(36,42)(38,41)(39,46)(40,45)(43,47)(44,48);(1,5)(2,10)(3,9)(4,12)(6,14)(7,16)(8,15)(11,13)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,6,7,3)(2,8,11,12)(4,13,15,10)(5,9,16,14)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,41)(35,39)(36,43)(38,46)(40,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8602395);