Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2270 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)(5,6)(7,10,15,13)(8,11)(9,12)(14,17,20,18)(16,19);(5,7)(6,10,12,13)(8,14,16,20)(9,15)(11,18)(17,19);(5,6)(7,13)(8,11)(9,12)(10,15)(14,18)(16,19)(17,20);(2,4)(5,8)(6,11)(7,14,15,20)(9,16)(10,17,13,18)(12,19)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225630, #72225637, #72225644, … |
| maximal normal subgroup of | #72225630, #72225637, #72225644, … |
| maximal quotient of | #72251452, #72265817, #72265818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337185);