Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,28)(22,30)(23,32)(25,31)(27,34)(29,33)(35,38)(36,40)(37,39);(1,2,3,4)(6,8)(9,11,13,24,15,23,14,25)(10,16,19,21,20,31,18,32)(12,22,27,37,29,36,28,38)(17,30,33,35,34,40,26,39);(9,12,14,28,15,29,13,27)(10,17,18,26,20,34,19,33)(11,22,25,38,23,36,24,37)(16,30,32,39,31,40,21,35);(11,23)(16,31)(21,32)(22,36)(24,25)(30,40)(35,39)(37,38);(9,13,15,14)(10,18,20,19)(11,24,23,25)(12,27,29,28)(16,32,31,21)(17,26,34,33)(22,37,36,38)(30,39,40,35)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72343604, #72343609, #72343610, … |
| maximal normal subgroup of | #72343604, #72343609, #72343610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279192);