Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361928 |
| 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 | #61410 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(4,5)(8,9)(10,15)(11,16)(12,14)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(1,2)(3,11)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,11)(15,16)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(1,3)(2,9)(4,12)(5,14)(6,13)(7,8)(10,16)(11,15)(18,25)(23,30)(24,32)(27,31);(1,4,6,5)(2,7,10,15)(3,12,13,14)(8,16,11,9)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283751 |
| maximal normal subgroup of | #72283751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269175);