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 | |
4498 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)(3,8)(4,6)(5,7)(10,15)(11,17)(12,19)(20,24)(25,26,30,32)(27,33,35,29)(28,31,37,38)(34,39,40,36);(3,4)(6,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27,30,35)(26,29,32,33)(28,34,37,40)(31,36,38,39);(1,3,5,4)(2,6,7,8)(25,26)(27,29)(28,31)(30,32)(33,35)(34,36)(37,38)(39,40);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,26)(27,33)(28,31)(29,35)(30,32)(34,39)(36,40)(37,38);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,37)(26,31,32,38)(27,34,35,40)(29,36,33,39);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29,30,33)(26,27,32,35)(28,36,37,39)(31,34,38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=hi,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, 8378362);