Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,14,15,12,16,13,7)(17,18)(19,25)(20,23,30,32)(21,27)(22,24)(26,31,28,29);(1,3,6,12)(2,7)(4,13,5,14)(8,16)(9,11)(10,15)(17,19,25,18,22,27,21,24)(20,28,29,32,30,26,31,23);(3,12)(7,15)(11,16)(13,14)(17,20)(18,23)(19,26)(21,31)(22,30)(24,32)(25,29)(27,28);(1,4,6,5)(2,8,10,9)(3,13,12,14)(7,16,15,11)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,f^b=fi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72302157, #72302206, #72305973, … |
| maximal normal subgroup of | #72302157, #72302206, #72305973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108807);