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 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,18)(11,13,17,15)(12,20,21,16)(19,23,24,22)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(3,8)(4,7)(12,21)(16,20)(19,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20);(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,23)(21,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(9,13)(10,17)(11,18)(12,22)(14,15)(16,24)(19,20)(21,23)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,37,38,39)Polycyclic presentation:
2^9|a^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8005000);