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 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,22,26)(19,25,21,23)(20,28,29,24)(27,31,30,32)(33,34)(35,44)(36,41)(37,47)(38,45)(39,43)(40,46)(42,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,29)(27,32)(30,31)(33,35)(34,40)(36,38)(37,45)(39,46)(41,42)(43,44)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,31,28,32)(33,36,39,47)(34,41,43,37)(35,38,46,48)(40,42,44,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,32)(26,28)(33,37)(34,36)(35,45)(38,40)(39,41)(42,46)(43,47)(44,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,21,22,19)(18,25,26,23)(20,27,29,30)(24,31,28,32)(33,34)(35,40)(36,41)(37,47)(38,42)(39,43)(44,46)(45,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,38,39,48)(34,42,43,45)(35,36,46,47)(37,40,41,44)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044241);