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 | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,14,12)(10,15,13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32)(33,34,40,44)(35,42,36,41)(37,43,39,38)(45,48,47,46);(1,3)(2,7)(4,11)(5,13)(6,9)(8,16)(10,14)(12,15)(33,35,40,36)(34,41,44,42)(37,47,39,45)(38,46,43,48);(3,9)(7,11)(10,13)(15,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,36,40,35)(34,42,44,41)(37,47,39,45)(38,46,43,48);(1,4,6,2)(3,7,9,11)(5,12,14,8)(10,15,13,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,37,40,39)(34,38,44,43)(35,45,36,47)(41,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,16)(13,14)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,38,40,43)(34,37,44,39)(35,46,36,48)(41,47,42,45);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,37)(34,43,44,38)(35,47,36,45)(41,46,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259616);