Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,25)(16,28)(17,22)(18,23)(19,24)(20,26)(21,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,16,19,27,18,26)(14,20,23,21,24,28,22,25);(2,4)(6,9)(10,12)(13,16,17,27,19,26,18,15)(14,21,23,28,24,25,22,20);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,26,27,16)(20,21,28,25);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459963);