Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10,13,17,15,18,14,30)(11,20,22,38,24,39,23,36)(12,25,28,35,27,37,29,19)(16,31,34,21,33,26,32,40);(9,11,15,24)(10,16,18,33)(12,26,27,31)(13,23,14,22)(17,32,30,34)(19,36,35,38)(20,25,39,37)(21,29,40,28);(1,3,4,6,5,7,8,2)(10,17,18,30)(12,27)(16,32,33,34)(19,37,35,25)(20,38,39,36)(21,40)(26,31)(28,29);(10,18)(16,33)(17,30)(19,35)(20,39)(25,37)(32,34)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287687, #72287729, #72287755, … |
| maximal normal subgroup of | #72287687, #72287729, #72287755, … |
| maximal quotient of | #72343639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391000);