Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1623 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)(3,12)(4,9)(5,10)(6,8)(7,11)(13,14)(15,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(2,8)(3,9)(4,14)(5,6)(10,11)(12,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3,7,9)(2,4,11,12)(5,15,6,13)(8,16,10,14)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4)(2,9)(3,11)(5,14)(6,16)(7,12)(8,15)(10,13);(1,5,7,6)(2,8,11,10)(3,13,9,15)(4,14,12,16)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72339754, #72339786, #72339813, … |
| maximal normal subgroup of | #72339754, #72339786, #72339813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272548);