Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3510 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)(6,9)(10,12)(13,14,17,16)(15,19,20,18)(22,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22,25,26)(23,27,28,24);(7,11)(10,12)(14,16)(18,19)(22,26)(23,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,19,20,18)(21,23)(22,27)(24,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 423746);