Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,14,17,19,16,15,18,20);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,15)(19,20);(7,12)(9,11);(6,10)(9,11);(13,15,17,20,16,14,18,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,d^a=dfi,c^b=cfi,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72313117, #72313124, #72313129, … |
| maximal normal subgroup of | #72313117, #72313124, #72313129, … |
| maximal quotient of | #72314188, #72314354, #72315194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277600);