Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368694 |
| 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 | |
1573 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)(3,9)(4,6)(5,14)(7,11)(8,15)(10,13)(12,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,6,12)(2,7,4,13)(5,10,15,11)(8,16,14,9)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,4,6,2)(3,7,12,13)(5,14,15,8)(9,10,16,11)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(3,10)(5,15)(7,16)(8,14)(9,13)(11,12)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(2,4)(7,13)(8,14)(9,16)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^9|b^2=h,c^2=hi,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=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306633, #72306650, #72306654, … |
| maximal normal subgroup of | #72306633, #72306650, #72306654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275928);