Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2087 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)(6,11)(7,15)(8,9)(12,20)(13,14)(17,18);(1,2,3,4)(5,6,8,11,10,13,9,14)(7,16,17,20,18,19,15,12);(6,11,13,14)(12,19,20,16);(2,4)(5,7)(6,12)(8,15)(9,17)(10,18)(11,19)(13,20)(14,16)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252190, #72256179, #72256189, … |
| maximal normal subgroup of | #72252190, #72256179, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330125);