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 | #45642 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,12)(10,16)(14,15)(17,18,23,24)(19,25,29,32)(20,30,31,21)(22,28,27,26)(33,34,38,39)(35,43,46,40)(36,45,47,41)(37,44,48,42);(2,4)(7,9)(8,12)(14,15)(18,24)(21,30)(25,32)(26,28)(34,39)(36,47)(40,43)(42,44);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,19,23,29)(18,25,24,32)(20,27,31,22)(21,28,30,26)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20)(18,21)(19,27)(22,29)(23,31)(24,30)(25,28)(26,32)(33,36,38,47)(34,41,39,45)(35,44,46,42)(37,40,48,43);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21)(18,20)(19,28)(22,32)(23,30)(24,31)(25,27)(26,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,48)(45,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,27)(18,26,24,28)(19,20,29,31)(21,32,30,25)(33,37,38,48)(34,42,39,44)(35,45,46,41)(36,40,47,43)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017820);