Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490485 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1667 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,8)(5,17)(10,20)(11,28)(13,14)(15,29)(16,30)(18,23)(19,25)(21,22)(26,27)(31,32);(1,2,8,22,5,11,18,20,6,10,23,28,17,21,4,7)(3,9,12,25,14,26,29,31,16,24,30,32,15,27,13,19);(1,3)(2,9)(4,13)(5,14)(6,16)(7,19)(8,12)(10,24)(11,26)(15,17)(18,29)(20,31)(21,27)(22,25)(23,30)(28,32);(2,10)(7,20)(9,24)(11,21)(19,31)(22,28)(25,32)(26,27)Polycyclic presentation:
2^9|b^2=eg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,f^c=fh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214066, #72214134, #72214308, … |
| maximal normal subgroup of | #72214066, #72214134, #72214308, … |
| maximal quotient of | #72216405, #72226932, #72226933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397681);