Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,15,17,16,18,19,20)(21,22,24,27,26,23,25,28);(2,4)(5,7,9,8)(6,10,11,12)(21,23)(22,26)(24,27)(25,28);(2,4)(5,8,9,7)(6,10,11,12)(22,27)(23,28)(24,25)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=eh,f^c=fh,g^c=gh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28821);