Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#418 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1490 |
| prime radical | #3584 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,7,14,25,29,8,11,26,30,6,27,24,12)(3,15,17,4,18,21,31,22,19,32,28,20,10,23,16,9);(1,3)(2,9)(4,14)(5,17)(6,10)(7,18)(8,19)(11,22)(12,23)(13,15)(16,24)(20,27)(21,29)(25,31)(26,28)(30,32);(1,4)(2,10)(3,14)(5,23)(6,9)(7,22)(8,20)(11,18)(12,17)(13,31)(15,25)(16,30)(19,27)(21,26)(24,32)(28,29);(2,11)(4,20)(9,22)(12,29)(13,30)(14,27)(15,32)(21,23)Polycyclic presentation:
2^9|a^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215848, #72215864, #72219678, … |
| maximal normal subgroup of | #72215848, #72215864, #72219678, … |
| maximal quotient of | C2×#481290 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388486);