Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,12)(10,11)(14,18)(16,24)(19,25)(20,27)(21,28)(22,26);(5,7,10,9)(6,8,11,12)(13,14,17,19)(15,21,23,27)(16,25,24,18)(20,26,28,22);(5,8,7,6,10,12,9,11)(14,19)(18,25)(20,28)(21,27);(13,15,17,23)(14,20,19,28)(16,26,24,22)(18,27,25,21)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=fi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 133311);