Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2262 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)(3,4)(9,10,14,19,15,18,13,28)(11,16,22,32,21,33,29,31)(12,23,26,36,27,17,25,35)(20,30,24,39,37,40,38,34);(2,4)(5,6,7,8)(9,11)(10,16)(12,24)(13,29)(14,22)(15,21)(17,34)(18,33)(19,32)(20,25)(23,39)(26,37)(27,38)(28,31)(30,35)(36,40);(6,8)(9,12,15,27)(10,17,18,23)(11,20,21,37)(13,26,14,25)(16,30,33,40)(19,36,28,35)(22,38,29,24)(31,39,32,34);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273790);