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 | #51965 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,19,23,20)(18,24,25,26)(21,30,32,28)(22,29,27,31)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(4,12)(8,13)(10,16)(14,15)(18,25)(22,27)(24,26)(29,31)(34,39)(37,45)(40,41)(43,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,19)(18,26,25,24)(21,28,32,30)(22,31,27,29)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21)(18,27)(19,28)(20,30)(22,25)(23,32)(24,29)(26,31)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,22,23,27)(18,21,25,32)(19,29,20,31)(24,30,26,28)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)Polycyclic presentation:
2^9|b^2=gi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300851);