Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2431 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3312 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)(7,12)(8,10)(9,11)(13,14)(15,18)(16,20)(17,19)(22,26)(23,24);(5,7,10,9)(6,11,8,12)(13,15,17,20)(14,18,19,16)(21,22)(23,28)(24,27)(25,26);(7,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23,25,24)(22,27,26,28);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,15)(14,16)(17,20)(18,19)(21,24,25,23)(22,27,26,28);(5,9,10,7)(6,11,8,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,28)(24,27)(25,26)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2036663);