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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3994 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)(3,10)(9,14)(11,16)(17,18,23,26)(19,24,27,22)(20,21,25,32)(28,29,30,31)(34,39)(35,44)(37,42)(45,47);(1,2,6,7)(3,4,10,8)(5,9,13,14)(11,12,16,15)(17,19,23,27)(18,24,26,22)(20,30,25,28)(21,31,32,29)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,48)(46,47);(3,10)(4,8)(5,13)(9,14)(19,27)(20,25)(21,32)(22,24)(33,35,38,44)(34,40,39,41)(36,47,43,45)(37,48,42,46);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,20)(18,21)(19,28)(22,31)(23,25)(24,29)(26,32)(27,30)(33,35)(34,41)(36,47)(37,46)(38,44)(39,40)(42,48)(43,45);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,25)(19,29)(20,26)(22,28)(23,32)(24,30)(27,31)(33,36,38,43)(34,42,39,37)(35,45,44,47)(40,48,41,46);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,19)(20,31)(21,28)(23,24)(25,29)(26,27)(30,32)(33,37,38,42)(34,43,39,36)(35,46,44,48)(40,47,41,45)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8969363);