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 | |
4012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,5,11,7)(4,12,14,8)(10,15,16,13)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31)(34,36)(38,40)(41,44)(45,47);(1,3,6,11)(2,7,9,5)(4,10,14,16)(8,15,12,13)(19,26)(20,25)(21,30)(23,29)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,5,6,7)(2,3,9,11)(4,13,14,15)(8,10,12,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,13)(14,16)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)(33,37,39,46)(34,41,36,44)(35,42,43,48)(38,45,40,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=g,e^2=gh,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, 8015918);