Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
912 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)(3,7,11,14,12,15,13,16)(17,18)(19,27)(20,24)(21,25,30,31)(22,26)(23,28,32,29);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(17,19,22,20)(18,23)(21,28,30,29)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16);(17,20,22,19)(18,24,26,27)(21,29,30,28)(23,31,32,25)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,f^b=fgi,f^e=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387737);