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 | #50807 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,8)(5,13)(12,16)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31)(34,40)(35,36)(42,47)(43,44);(3,8)(4,11)(10,15)(12,16)(18,24)(20,30)(23,26)(27,29)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3,6,8)(2,4,7,11)(5,10,13,15)(9,12,14,16)(17,20,22,30)(18,25,24,28)(19,27,21,29)(23,31,26,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,30)(29,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21,22,19)(18,26,24,23)(20,29,30,27)(25,32,28,31)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034764);