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 | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,23)(21,22)(25,26)(27,29)(28,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,19)(18,20)(21,24)(22,23)(26,31)(27,28);(2,4)(6,7)(10,12)(14,15)(18,21)(19,23)(25,27,30,28)(26,29,31,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,20,22,24)(18,23,21,19)(25,28,30,27)(26,29,31,32);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,24,23,20)(25,29)(26,27)(28,31)(30,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=fg,d^2=h,e^2=g,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.
| maximal subgroup of | #72334169, #72334195, #72334197, … |
| maximal normal subgroup of | #72334169, #72334195, #72334197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2056109);