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 | C2×#575 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,12,20,27,7,9,22,26,21,11,4,18)(3,13,14,8,16,24,31,25,17,29,30,23,15,32,19,28);(1,3)(2,8)(4,19)(5,14)(6,16)(7,17)(9,23)(10,24)(11,28)(12,25)(13,18)(15,21)(20,31)(22,30)(26,32)(27,29);(4,20)(5,22)(10,26)(13,29)(14,30)(18,27)(19,31)(24,32);(2,9)(8,23)(10,26)(11,12)(13,29)(18,27)(24,32)(25,28)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,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214208, #72214259, #72214311, … |
| maximal normal subgroup of | #72214208, #72214259, #72214311, … |
| maximal quotient of | #72219605, #72224803 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388484);