Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,11,14)(7,9,12,10)(8,15)(13,17)(16,20)(18,19);(1,2)(3,4)(5,7,9,14,11,12,10,6)(8,15,17,20,18,19,16,13);(2,4)(6,12)(7,14)(8,16)(9,10)(13,20)(17,18);(5,8)(6,13)(7,15)(9,17)(10,16)(11,18)(12,19)(14,20)Polycyclic presentation:
2^9|a^2=ef,b^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238735 |
| maximal normal subgroup of | #72238735 |
| maximal quotient of | #72238740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363706);