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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(18,23)(21,29)(24,25)(27,28)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34,38,40)(35,36,43,41)(37,39,46,45)(42,44,48,47);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,31)(29,30)(33,36,38,41)(34,35,40,43)(37,44,46,47)(39,42,45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,22,19)(18,25,23,24)(21,28,29,27)(26,32,30,31)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8079819);