Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 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)(6,8)(10,11)(13,14,17,15,19,22,18,20)(16,24,25,28,26,23,27,21);(5,6,9,8)(7,11,12,10)(13,15,18,14,19,20,17,22)(16,23,25,21,26,24,27,28);(2,4)(5,7,9,12)(6,10,8,11)(14,20)(15,22)(17,18)(21,24)(23,28)(25,27);(5,8,9,6)(7,10,12,11)(13,16)(14,21)(15,23)(17,27)(18,25)(19,26)(20,24)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372084);