Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,13,10,11,9,14)(7,15,18,19,17,20,16,12);(1,2)(3,4)(6,11)(7,16)(8,9)(12,15)(17,18)(19,20);(2,4)(5,7)(6,12)(8,18)(9,16)(10,17)(11,19)(13,15)(14,20);(6,13)(8,9)(11,14)(12,15)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284694, #72284893, #72285507, … |
| maximal normal subgroup of | #72284694, #72284893, #72285507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326059);