Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,9,13,10,12,15,16)(17,18,20,27,22,25,21,24)(19,28,29,32,30,23,31,26);(18,23)(19,29)(20,21)(24,26)(25,28)(27,32)(30,31);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(18,24,25,27)(19,30)(23,26,28,32)(29,31);(2,7)(6,12)(8,14)(13,16)(18,25)(23,28)(24,27)(26,32)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216779, #72216788, #72216906, … |
| maximal normal subgroup of | #72216779, #72216788, #72216906, … |
| maximal quotient of | #72285836, #72285943, #72286037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391069);