Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 | |
1026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,25)(17,22)(18,23)(20,24)(21,26)(27,28);(5,7,9,12)(6,10,8,11)(13,15,19,14,20,23,18,24)(16,22,26,28,27,21,17,25);(1,2,4,3)(6,8)(10,11)(13,16)(14,21)(15,25)(17,19)(18,26)(20,27)(22,24)(23,28);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,17,19,16,20,26,18,27)(14,22,23,28,24,21,15,25)Polycyclic presentation:
2^9|a^2=eg,b^2=ei,c^2=ef,d^2=f,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^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370435);