Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×H(5,𝔽2) | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,14,18)(11,15,20,28)(12,22,24,37)(13,17,26,33)(16,30,31,23)(19,34,35,40)(21,29,25,32)(27,38,39,36);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,25)(14,20)(16,27)(17,32)(18,28)(21,26)(22,34)(23,36)(24,35)(29,33)(30,38)(31,39)(37,40);(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,22)(18,31)(20,35)(21,36)(25,38)(26,30)(28,39)(29,34)(32,40)(33,37);(1,4,5,3)(2,7,8,6)(11,20)(15,28)(19,35)(21,25)(27,39)(29,32)(34,40)(36,38)Polycyclic presentation:
2^9|a^2=ei,b^2=e,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408182);