Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,16)(9,13)(10,14)(12,18)(15,20)(17,19);(1,3,4,2)(5,7)(6,10)(8,17)(9,15)(11,16)(12,20)(13,19)(14,18);(2,3)(5,7)(6,11)(8,17)(9,15)(10,12)(13,20)(14,18)(16,19);(5,6)(7,10)(8,16)(9,13)(11,17)(12,18)(14,20)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363576);