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×#127 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,8,11,12)(4,9,13,15)(5,10,14,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,3)(2,8)(4,5)(6,12)(7,11)(9,10)(13,14)(15,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,47)(44,48)(45,46);(3,11)(4,13)(8,12)(9,15)(20,29)(24,28)(27,30)(31,32)(33,34,37,41)(35,39,43,38)(36,40,46,44)(42,48,45,47);(1,4)(2,9)(3,5)(6,15)(7,13)(8,10)(11,14)(12,16)(17,20,22,29)(18,24,25,28)(19,27,21,30)(23,31,26,32)(33,34)(35,39)(36,44)(37,41)(38,43)(40,46)(42,47)(45,48);(1,5)(2,10)(3,4)(6,16)(7,14)(8,9)(11,13)(12,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,29)(30,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,47)(42,46)(44,48);(1,6,7,2)(3,12,11,8)(4,9,13,15)(5,10,14,16)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8629420);