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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(18,20)(23,26)(24,28)(30,32)(33,34,39,43)(35,40,46,44)(36,38,42,48)(37,47,45,41);(2,8)(3,4)(10,15)(11,12)(19,25)(23,26)(27,31)(30,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,39,46)(34,40,43,44)(36,45,42,37)(38,41,48,47);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,38)(35,45)(37,46)(39,42)(40,41)(43,48)(44,47);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=ghi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8365320);