Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #61316 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,3,10,8)(4,11,15,12,14,7,13,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3)(2,6)(4,12)(5,8)(7,15)(9,10)(11,13)(14,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(2,3)(7,11)(8,9)(12,16)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,4)(2,7)(3,11)(5,15)(6,14)(8,12)(9,16)(10,13)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287738 |
| maximal normal subgroup of | #72287738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128233);