Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
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 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,19)(13,20);(5,6)(7,13)(8,16)(9,11)(10,15)(12,18)(14,20)(17,19);(5,7)(6,10)(8,17)(9,14)(11,19)(12,20)(13,16)(15,18);(2,4)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214896, #72224751, #72226719, … |
| maximal normal subgroup of | #72214896, #72224751, #72226719, … |
| maximal quotient of | #72213205, #72213848, #72214321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301132);