Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20);(1,3,4,2)(6,9)(10,12)(21,22,24,26)(23,27,28,25);(14,17)(18,20)(21,23,24,28)(22,25,26,27);(6,9)(10,12)(13,15,16,19)(14,18,17,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421575);