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 | |
4658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,34)(28,31)(29,33)(30,32)(35,40)(36,37)(38,39);(2,3)(6,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,3,5,2)(4,7,8,6)(10,15)(11,18)(16,21)(19,24)(26,28)(32,34)(33,36)(39,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28,31,26)(27,34,30,32)(29,36,37,33)(35,40,38,39);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30,31,27)(26,34,28,32)(29,35,37,38)(33,39,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9722067);