Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3518 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)(9,10,15,30,14,20,13,19)(11,17,28,16,22,34,23,32)(12,24,29,37,27,36,26,18)(21,38,40,35,39,31,25,33);(5,6)(7,8)(9,11)(10,16)(12,25)(13,28)(14,22)(15,23)(17,30)(18,35)(19,34)(20,32)(21,26)(24,38)(27,40)(29,39)(31,36)(33,37);(2,4)(6,8)(9,11)(10,17)(12,21)(13,23)(14,22)(15,28)(16,30)(18,33)(19,32)(20,34)(24,38)(25,26)(27,39)(29,40)(31,36)(35,37);(2,4)(9,12,14,27)(10,18,20,37)(11,21,22,39)(13,29,15,26)(16,31,32,38)(17,33,34,35)(19,24,30,36)(23,40,28,25);(11,22)(16,32)(17,34)(21,39)(23,28)(25,40)(31,38)(33,35)Polycyclic presentation:
2^9|a^2=fgi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410554);