Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,18)(11,19,20,13)(12,16,26,32)(14,28,29,17)(21,30,23,24)(22,27,34,33)(25,38,39,31)(35,37,36,40);(5,6,7,8)(11,20)(13,19)(21,23)(22,34)(24,30)(27,33)(35,36)(37,40);(1,2)(3,4)(9,11)(10,13)(12,23)(14,22)(15,20)(16,30)(17,27)(18,19)(21,26)(24,32)(25,36)(28,33)(29,34)(31,40)(35,39)(37,38);(9,12)(10,16)(11,21)(13,24)(14,25)(15,26)(17,31)(18,32)(19,30)(20,23)(22,35)(27,37)(28,38)(29,39)(33,40)(34,36);(9,13)(10,11)(12,24)(14,27)(15,19)(16,21)(17,22)(18,20)(23,32)(25,37)(26,30)(28,34)(29,33)(31,35)(36,38)(39,40);(9,14,15,29)(10,17,18,28)(11,22,20,34)(12,25,26,39)(13,27,19,33)(16,31,32,38)(21,35,23,36)(24,37,30,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537852);