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 | #2404 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4123 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)(8,13)(10,16)(14,15)(17,18,23,21)(19,22,27,24)(20,28,29,25)(26,31,32,30)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,27)(18,24,21,22)(20,26,29,32)(25,31,28,30)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(2,6)(7,9)(8,13)(14,15)(20,29)(25,28)(26,32)(30,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,25)(19,26)(21,28)(22,30)(23,29)(24,31)(27,32)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,18)(19,22,27,24)(20,25,29,28)(26,31,32,30)(33,37,39,35)(34,36,42,40)(38,45,47,43)(41,44,46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,19)(20,30)(21,27)(23,24)(25,26)(28,32)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=g,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, 8062212);