Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,15)(8,18)(9,13)(10,14)(11,16)(12,19)(17,20);(5,7)(6,11)(8,9)(10,17)(12,20)(13,15)(14,18)(16,19);(1,2,3,4)(5,6)(7,15)(8,11)(9,12)(10,14)(13,19)(16,18)(17,20);(5,8,19,17,10,16,9,7)(6,11,13,20,14,18,12,15)Polycyclic presentation:
2^9|c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213250, #72213603, #72215938, … |
| maximal normal subgroup of | #72213250, #72213603, #72215938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330806);