Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,8,10,11,7,12)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(5,6)(7,8)(9,12)(10,11)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26);(1,2,3,4)(5,7,10,9)(6,8,11,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(5,8,7,6,10,12,9,11)(16,23)(19,21)(24,28)(26,27);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=gh,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105779);