Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3356 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)(7,13)(8,16)(9,12)(10,14)(11,17)(15,19)(18,20);(5,7,8,14,9,15,17,20)(6,10,11,13,12,18,16,19);(10,18)(13,19);(2,4)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72214576, #72216067, … |
| maximal normal subgroup of | #72214076, #72214576, #72216067, … |
| maximal quotient of | C2×#394195, #72236790, #72237212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301391);