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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,17,20,18,22,15,19)(16,21,25,27,24,28,23,26);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,19)(15,17)(16,24)(20,22)(21,27)(26,28);(1,3,4,2)(6,9)(10,11)(13,15,18,17)(14,20,22,19)(16,23,24,25)(21,27,28,26);(5,7,8,12)(6,10,9,11)(13,16,18,24)(14,21,22,28)(15,23,17,25)(19,26,20,27);(16,24)(21,28)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=fgi,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, 6371302);