Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2512 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)(7,8)(9,11)(10,12)(13,14,17,20)(15,16,19,18)(22,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,20)(17,19)(22,26)(27,28);(5,8,10,9)(6,11,12,7)(13,14,17,20)(15,16,19,18)(21,22)(23,27)(24,28)(25,26);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23,25,24)(22,27,26,28);(5,9,10,8)(6,11,12,7)(13,16)(14,19)(15,20)(17,18)(21,24,25,23)(22,28,26,27)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 425255);