Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,3,5,11)(4,10,13,16,14,8,15,12)(18,24)(19,27)(20,29)(21,22)(25,28)(26,31);(1,3,7,2)(4,12,14,16)(5,9,6,11)(8,13,10,15)(17,18,23,26)(19,20,28,29)(21,31,22,24)(25,32,27,30);(17,19,22,27,23,28,21,25)(18,20,24,30,26,29,31,32);(1,4)(2,8)(3,10)(5,15)(6,13)(7,14)(9,12)(11,16)(17,20)(18,25)(19,24)(21,32)(22,30)(23,29)(26,27)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,e^d=eh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72286992, #72287015, #72287034, … |
| maximal normal subgroup of | #72286992, #72287015, #72287034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132979);