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 | |
3822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,41,45,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,48,47);(3,11)(7,8)(12,16)(13,14)(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,48,47);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,47,43,41);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,34)(35,39)(36,40)(37,47)(38,42)(41,45)(43,44)(46,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,37)(34,41)(35,36)(38,45)(39,48)(40,43)(42,47)(44,46)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641111);