Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,20,24,23,27,32,26)(19,22,28,31,25,21,29,30);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,22,24,20,29,31,18,23,25,21,26,32,28,30,27);(3,11)(7,15)(10,16)(12,13)(17,20,23,32)(18,24,27,26)(19,28,25,29)(21,30,22,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=ch,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321750, #72321765, #72321794, … |
| maximal normal subgroup of | #72321750, #72321765, #72321794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34035);