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 | |
1834 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)(6,7)(9,10,14,18,15,19,13,27)(11,16,21,32,22,33,28,31)(12,23,26,17,25,36,29,35)(20,34,38,39,37,40,24,30);(5,6,8,7)(9,11)(10,16)(12,24)(13,28)(14,21)(15,22)(17,34)(18,32)(19,33)(20,26)(23,30)(25,38)(27,31)(29,37)(35,40)(36,39);(2,4)(5,7,8,6)(9,12)(10,17)(11,20)(13,29)(14,26)(15,25)(16,30)(18,36)(19,35)(21,38)(22,37)(23,27)(24,28)(31,40)(32,34)(33,39);(12,25)(17,35)(20,37)(23,36)(24,38)(26,29)(30,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265292);