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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(17,18,23,20)(19,22,27,24)(21,25,30,28)(26,29,32,31)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,11)(2,8,9,16)(4,12,14,5)(6,10,15,13)(17,18,23,20)(19,24,27,22)(21,28,30,25)(26,29,32,31)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,9)(3,11)(4,14)(10,13)(17,19,23,27)(18,24,20,22)(21,26,30,32)(25,31,28,29)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,20,23,18)(19,22,27,24)(21,28,30,25)(26,29,32,31)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,21,23,30)(18,25,20,28)(19,26,27,32)(22,29,24,31)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,22)(18,19)(20,27)(21,29)(23,24)(25,26)(28,32)(30,31)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662649);