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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3840 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(19,20)(23,26)(27,28)(31,32)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(2,4)(3,10)(9,12)(11,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,18)(19,23)(20,26)(21,29)(22,25)(24,30)(27,32)(28,31)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227571);