Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,8)(7,11,12,10)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(5,7,9,12)(6,10,8,11)(13,14,16,19,17,20,26,24)(15,21,22,27,23,28,25,18);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21);(7,12)(10,11)(13,14,17,20)(15,18,23,27)(16,24,26,19)(21,22,28,25)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341290);