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 | |
4036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(18,23)(20,27)(21,29)(26,31)(33,34,39,42)(35,40,37,36)(38,41,46,48)(43,47,45,44);(2,6)(7,12)(8,13)(14,16)(18,23)(19,25)(20,27)(28,32)(33,35,39,37)(34,36,42,40)(38,45,46,43)(41,47,48,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,37,39,35)(34,40,42,36)(38,45,46,43)(41,47,48,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,38,39,46)(34,41,42,48)(35,43,37,45)(36,44,40,47);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8016368);