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 | #45676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4052 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,22,23)(19,27,28,24)(20,25,30,32)(21,31,29,26)(33,34,39,42)(35,38,45,40)(36,41,46,37)(43,44,47,48);(2,6)(7,9)(8,12)(14,15)(18,23)(19,28)(20,30)(26,31)(36,46)(37,41)(43,47)(44,48);(17,19,22,28)(18,24,23,27)(20,29,30,21)(25,31,32,26)(33,35,39,45)(34,40,42,38)(36,43,46,47)(37,44,41,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,31)(30,32)(33,36)(34,41)(35,43)(37,42)(38,44)(39,46)(40,48)(45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,25)(19,29)(21,28)(22,30)(23,32)(24,31)(26,27)(33,37,39,41)(34,36,42,46)(35,44,45,48)(38,47,40,43);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21,22,29)(18,26,23,31)(19,20,28,30)(24,25,27,32)(33,38)(34,35)(36,44)(37,47)(39,40)(41,43)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000509);