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 | |
3816 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,22,27)(19,25,28,23)(20,29,30,24)(21,31,32,26)(33,34,39,37)(35,42,38,40)(36,41,45,44)(43,48,46,47);(3,9)(6,13)(10,15)(14,16)(19,28)(20,30)(23,25)(24,29)(33,35,39,38)(34,40,37,42)(36,43,45,46)(41,47,44,48);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,28)(18,23,27,25)(20,31,30,26)(21,24,32,29)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,22,30)(18,24,27,29)(19,26,28,31)(21,25,32,23)(33,37,39,34)(35,40,38,42)(36,44,45,41)(43,47,46,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,25)(20,26)(21,29)(22,28)(23,27)(24,32)(30,31)(33,38,39,35)(34,40,37,42)(36,43,45,46)(41,48,44,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21,22,32)(18,26,27,31)(19,24,28,29)(20,25,30,23)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947687);