Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2665 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)(15,19)(16,20)(17,18);(1,2)(3,4)(6,9)(10,12)(13,15,16,17)(14,18,20,19);(2,4)(14,18)(15,17)(19,20);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,16,17)(14,19,20,18);(2,4)(14,20)(18,19)Polycyclic presentation:
2^9|b^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72257557, #72257580, #72257619, … |
| maximal normal subgroup of | #72257557, #72257580, #72257619, … |
| maximal quotient of | #72305528, #72305709, #72310182 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482610);