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 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(4,12)(13,15)(20,27)(24,29)(26,31)(30,32)(34,39)(35,37)(41,44)(43,45);(1,2,6,7)(3,5,10,8)(4,9,12,14)(11,13,16,15)(18,22)(19,25)(24,29)(26,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,48)(45,47);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,46)(34,41,39,44)(35,43,37,45)(40,47,42,48);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=gh,e^2=gh,f^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, 8015979);