Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4920 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)(7,10)(8,9)(11,12)(13,14)(15,17)(16,19)(18,20);(1,2)(3,4)(14,17)(15,20)(18,19);(2,4)(6,9)(10,12);(2,4);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237257, #72239702, #72239790, … |
| maximal normal subgroup of | #72237257, #72239702, #72239790, … |
| maximal quotient of | #72237276, #72305806, #72311557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360905);