Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(9,10,14,19)(11,21,25,16)(12,26,29,36)(13,20,15,17)(18,30,35,28)(22,33,24,32)(23,39,40,34)(27,37,38,31);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,27)(13,22)(14,25)(15,24)(17,33)(18,34)(19,21)(20,32)(23,28)(26,31)(29,38)(30,40)(35,39)(36,37);(10,17)(11,22)(13,15)(16,21)(18,26)(19,20)(23,38)(24,25)(27,40)(28,30)(31,37)(35,36);(6,8)(9,12,14,29)(10,18,19,35)(11,23,25,40)(13,28,15,30)(16,31,21,37)(17,26,20,36)(22,38,24,27)(32,34,33,39);(1,3,4,2)(9,13,14,15)(10,17,19,20)(11,24,25,22)(12,28,29,30)(16,32,21,33)(18,26,35,36)(23,27,40,38)(31,34,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426481);