Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,18,19,17,22,15,21)(16,20,25,26,24,28,23,27);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,19)(15,18)(16,24)(20,27)(21,22)(26,28);(1,3,4,2)(6,9)(10,11)(13,15,17,18)(14,19,22,21)(16,23,24,25)(20,26,28,27);(5,7,8,12)(6,10,9,11)(13,16,17,24)(14,20,22,28)(15,23,18,25)(19,26,21,27);(16,24)(20,28)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371300);