Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(9,10,15,19)(11,20,23,31)(12,17,26,35)(13,18,14,29)(16,27,32,22)(21,37,24,40)(25,34,28,36)(30,38,33,39);(5,6)(7,8)(9,11,13,27,15,23,14,22)(10,16,29,31,19,32,18,20)(12,24,28,39,26,21,25,38)(17,33,34,37,35,30,36,40);(2,4)(6,8)(9,12)(10,17)(11,21)(13,28)(14,25)(15,26)(16,30)(18,36)(19,35)(20,37)(22,39)(23,24)(27,38)(29,34)(31,40)(32,33);(5,6)(7,8)(9,13,15,14)(10,18,19,29)(11,22,23,27)(12,25,26,28)(16,31,32,20)(17,34,35,36)(21,38,24,39)(30,37,33,40);(6,8)(11,23)(12,26)(16,32)(17,35)(20,31)(22,27)(25,28)(34,36)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481249);