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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,24,28)(19,27,22,23)(20,26,21,25)(29,32,30,31)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,19,24,22)(18,23,28,27)(20,29,21,30)(25,32,26,31)(34,39)(35,42)(41,44)(43,48);(2,7)(5,15)(8,13)(9,16)(17,20,24,21)(18,25,28,26)(19,29,22,30)(23,32,27,31)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,13)(11,14)(15,16)(17,21,24,20)(18,26,28,25)(19,29,22,30)(23,32,27,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,16)(8,15)(10,12)(17,22,24,19)(18,27,28,23)(20,30,21,29)(25,31,26,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48);(1,5,6,15)(2,10,7,11)(3,9,12,16)(4,13,14,8)(17,23,24,27)(18,19,28,22)(20,31,21,32)(25,30,26,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229063);