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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4217 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(2,7)(5,9)(8,10)(13,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(33,35,38,46)(34,39,42,43)(36,44,45,37)(40,41,47,48);(3,11)(8,10)(12,14)(13,16)(19,27)(20,29)(23,28)(24,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,36)(34,40)(35,44)(37,46)(38,45)(39,41)(42,47)(43,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746433);