Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362950 |
| 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 | #61461 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,9,12,16)(5,14,15,8)(7,11,10,13)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(1,3)(2,7)(4,10)(5,11)(6,12)(8,16)(9,14)(13,15);(1,4,6,2)(3,10,12,7)(5,14,15,8)(9,13,16,11)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13);(3,12)(7,10)(9,16)(11,13)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231256, C2×#6363041, #72306691, … |
| maximal normal subgroup of | #72231256, C2×#6363041, #72306691, … |
| maximal quotient of | #72280022, C2×#6363041, #72309438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6270237);