Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of C22⋊#2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C4⋊#2104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#401 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13,15,10)(8,11,17,20)(9,12)(14,19)(16,18);(5,7)(6,10)(8,16,17,14)(9,15)(11,19,20,18)(12,13);(5,8)(6,11,12,20)(7,14)(9,17)(10,18,13,19)(15,16);(1,2,3,4)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320923);