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 | #51169 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4138 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)(19,28)(20,22)(21,31)(23,25)(24,32)(26,30)(27,29)(33,34,39,41)(35,40,43,37)(36,42,45,47)(38,44,48,46);(1,2)(3,9)(4,11)(5,7)(6,8)(10,15)(12,16)(13,14)(17,19,23,32)(18,24,25,28)(20,29,30,21)(22,31,26,27)(33,35)(34,40)(36,38)(37,41)(39,43)(42,44)(45,48)(46,47);(5,13)(7,14)(10,16)(12,15)(18,25)(22,26)(24,28)(27,31)(34,41)(35,43)(38,48)(42,47);(1,3,6,11)(2,4,8,9)(5,10,13,16)(7,12,14,15)(17,20)(18,26)(19,29)(21,32)(22,25)(23,30)(24,27)(28,31)(33,36)(34,42)(35,38)(37,46)(39,45)(40,44)(41,47)(43,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,29)(18,27,25,31)(19,30,32,20)(22,24,26,28)(33,37,39,40)(34,43,41,35)(36,44,45,46)(38,47,48,42);(1,5,6,13)(2,7,8,14)(3,10,11,16)(4,12,9,15)(17,22,23,26)(18,20,25,30)(19,31,32,27)(21,24,29,28)(33,38)(34,44)(35,36)(37,47)(39,48)(40,42)(41,46)(43,45)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=ghi,e^2=h,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, 8300951);