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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10760 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)(3,4)(6,8)(13,26)(17,30)(21,35)(24,37)(28,32)(29,36)(33,40)(38,39);(5,6)(7,8)(9,10,14,18)(11,19,22,15)(12,16,25,23)(13,17,26,30)(20,31,34,27)(21,32,35,28)(24,29,37,36)(33,39,40,38);(2,4)(9,10)(11,15)(12,23)(13,17)(14,18)(16,25)(19,22)(20,31)(21,28)(24,36)(26,30)(27,34)(29,37)(32,35)(33,39)(38,40);(1,2)(3,4)(9,11,14,22)(10,15,18,19)(12,20,25,34)(13,21,26,35)(16,27,23,31)(17,28,30,32)(24,33,37,40)(29,38,36,39);(9,12)(10,16)(11,20)(13,24)(14,25)(15,27)(17,29)(18,23)(19,31)(21,33)(22,34)(26,37)(28,38)(30,36)(32,39)(35,40);(9,13)(10,17)(11,21)(12,24)(14,26)(15,28)(16,29)(18,30)(19,32)(20,33)(22,35)(23,36)(25,37)(27,38)(31,39)(34,40)Polycyclic presentation:
2^9|b^2=i,d^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72252227, #72316203 |
| maximal normal subgroup of | #72252227, #72316203 |
| maximal quotient of | #72252228 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7580732);