Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #288787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,7,10,12,8,11)(13,14)(15,23)(16,19)(17,26)(18,20)(21,24)(22,27)(25,28);(1,2)(3,4)(5,7,8,6,10,11,9,12)(13,15,17,16,18,24,27,25)(14,19,22,23,20,28,26,21);(2,4)(6,7)(8,9)(11,12)(14,20)(15,24)(16,25)(22,26);(13,16,17,24,18,25,27,15)(14,21,22,19,20,23,26,28)Polycyclic presentation:
2^9|a^2=fhi,b^2=efh,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 196024);