Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,10)(11,12)(14,15,21,23)(20,22,27,28);(5,6)(7,11)(8,12)(9,10)(13,14,18,15,19,21,17,23)(16,22,25,27,26,28,24,20);(5,7,9,8)(6,11,10,12)(13,15,18,14,19,23,17,21)(16,22,25,20,26,28,24,27);(2,4)(5,8,9,7)(6,11,10,12)(13,16,18,25,19,26,17,24)(14,20,23,28,21,27,15,22)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ei,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340489);