Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,16)(17,19);(5,6)(7,10,14,16)(8,11)(9,12)(13,19,20,17)(15,18);(5,7)(6,10,12,16)(8,13)(9,14)(11,17,18,19)(15,20);(2,4)(5,8)(6,11)(7,13)(9,15)(10,17)(12,18)(14,20)(16,19)Polycyclic presentation:
2^9|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225625, #72225636, #72225638, … |
| maximal normal subgroup of | #72225625, #72225636, #72225638, … |
| maximal quotient of | #72251465, #72251474, #72265715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329991);