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 | #2818 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3841 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)(3,8,13,14)(4,6,12,10)(5,15,16,11)(17,18,23,22)(19,24,29,28)(20,30,25,21)(26,32,31,27)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3)(2,8)(4,11)(5,6)(7,13)(9,14)(10,16)(12,15)(17,19,23,29)(18,24,22,28)(20,26,25,31)(21,27,30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,9)(3,13)(6,10)(11,15)(17,20)(18,21)(19,26)(22,30)(23,25)(24,27)(28,32)(29,31)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,7,12)(2,10,9,6)(3,11,13,15)(5,8,16,14)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,11)(3,10)(4,14)(6,13)(7,16)(8,12)(9,15)(17,21,23,30)(18,25,22,20)(19,27,29,32)(24,31,28,26)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,6,7,10)(2,12,9,4)(3,5,13,16)(8,15,14,11)(17,22,23,18)(19,28,29,24)(20,30,25,21)(26,32,31,27)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324512);