Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1859 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,20,17,21,16,22)(15,23,27,19,26,25,24,28);(5,6)(7,11)(8,10)(9,12)(13,15,17,26)(14,19,21,28)(16,27,18,24)(20,23,22,25);(2,4)(5,7,8,12)(6,9,10,11)(13,16,17,18)(14,20,21,22)(15,24,26,27)(19,25,28,23);(7,12)(9,11)(13,14,16,22,17,21,18,20)(15,25,24,19,26,23,27,28);(6,10)(9,11)(14,21)(15,26)(20,22)(24,27)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972179);