Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,8,7)(9,10,14,18,15,20,13,19)(11,21,24,16,23,33,25,32)(12,26,30,17,28,36,29,35)(22,31,27,40,38,39,37,34);(1,2)(3,4)(5,7,8,6)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,34)(18,33)(19,21)(20,32)(22,29)(26,39)(28,37)(30,38)(31,36)(35,40);(2,4)(6,7)(9,12,13,29,15,28,14,30)(10,17,18,36,20,35,19,26)(11,22,24,27,23,38,25,37)(16,31,21,34,32,39,33,40);(11,23)(12,28)(16,32)(17,35)(21,33)(24,25)(26,36)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265560);