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 | #53408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(18,21)(19,20)(22,30)(31,32)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,15)(13,14)(17,18,23,21)(19,24,20,27)(22,25,30,29)(26,31,28,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,5)(3,9)(8,13)(10,12)(17,19,23,20)(18,24,21,27)(22,26,30,28)(25,31,29,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,4,7,15)(2,8,5,13)(3,10,9,12)(6,14,11,16)(17,20,23,19)(18,24,21,27)(22,28,30,26)(25,31,29,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,5,7,2)(3,11,9,6)(4,13,15,8)(10,16,12,14)(17,21,23,18)(19,24,20,27)(22,29,30,25)(26,31,28,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6,7,11)(2,9,5,3)(4,14,15,16)(8,12,13,10)(17,22,23,30)(18,25,21,29)(19,26,20,28)(24,31,27,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027112);