Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,16,22)(11,23,27,35)(12,18,31,34)(13,32,24,37)(14,21,15,20)(17,26,36,25)(19,33,40,28)(29,39,30,38);(1,2)(3,4)(5,7,8,6)(9,11)(10,17)(12,28)(13,29)(14,26)(15,25)(16,27)(18,37)(19,39)(20,35)(21,23)(22,36)(24,30)(31,33)(32,34)(38,40);(6,7)(9,12)(10,18)(11,24)(13,27)(14,30)(15,29)(16,31)(17,19)(20,39)(21,38)(22,34)(23,37)(25,28)(26,33)(32,35)(36,40);(2,4)(6,7)(9,13,16,24)(10,19,22,40)(11,12,27,31)(14,33,15,28)(17,34,36,18)(20,37,21,32)(23,38,35,39)(25,30,26,29)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328050);