Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2594 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)(7,11)(8,9)(10,12)(13,14,17,16)(15,18,19,20)(22,23)(24,28);(6,9)(7,12)(14,16)(18,20)(22,23)(26,27);(6,9)(10,11)(15,19)(18,20)(21,22,25,23)(24,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20)(21,23,25,22)(24,27,28,26);(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,20,19,18)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 492841);