Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,16,20,18)(22,23)(24,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,18,19,16)(22,23)(26,27);(6,10)(9,11)(14,19)(15,20)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,16,17,18)(14,20,19,15)(21,23,25,22)(24,27,28,26);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=fg,d^2=h,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, 1993632);