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 | #51846 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18)(19,25)(20,24)(21,26)(22,23)(27,29)(28,30)(31,32)(33,34,39,41)(35,44,46,40)(36,42,47,37)(38,45,48,43);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,18)(19,21)(20,28)(22,23)(24,30)(25,26)(27,29)(31,32)(33,35,39,46)(34,40,41,44)(36,45,47,43)(37,48,42,38);(2,9)(3,13)(4,15)(11,16)(18,23)(19,26)(24,28)(27,32)(34,41)(36,47)(40,44)(43,45);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,31)(28,29)(30,32)(33,36)(34,42)(35,45)(37,41)(38,40)(39,47)(43,46)(44,48);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,20)(18,24)(19,27)(21,31)(22,30)(23,28)(25,29)(26,32)(33,37,39,42)(34,36,41,47)(35,38,46,48)(40,43,44,45);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,21)(18,19)(20,29)(22,25)(23,26)(24,32)(27,28)(30,31)(33,38)(34,43)(35,37)(36,40)(39,48)(41,45)(42,46)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9625775);