Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359664 |
| 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 |
| prime radical | #61590 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,11,15)(4,6,12,9)(10,16,13,14)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(2,6)(3,10)(4,12)(7,15)(8,9)(11,13)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(1,3,5,11)(2,7,8,15)(4,13,12,10)(6,14,9,16);(2,8)(6,9)(7,15)(14,16)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339700, #72339717, #72339778 |
| maximal normal subgroup of | #72339700, #72339717, #72339778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267093);