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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,11,4)(5,7,13,14)(10,15,16,12)(17,18,23,26)(19,24,27,22)(20,21,25,32)(28,29,30,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,2)(3,9)(4,11)(5,7)(6,8)(10,15)(12,16)(13,14)(17,19,23,27)(18,24,26,22)(20,30,25,28)(21,31,32,29)(34,35)(39,41)(40,42)(46,47);(5,13)(7,14)(10,16)(12,15)(19,27)(22,24)(28,30)(29,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3,6,11)(2,4,8,9)(5,10,13,16)(7,12,14,15)(17,20)(18,21)(19,28)(22,31)(23,25)(24,29)(26,32)(27,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21)(18,25)(19,29)(20,26)(22,28)(23,32)(24,30)(27,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,6,13)(2,7,8,14)(3,10,11,16)(4,12,9,15)(17,22)(18,19)(20,31)(21,28)(23,24)(25,29)(26,27)(30,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=g,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, 8296472);