Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499960 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23930 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
961 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)(3,11)(4,16)(5,21)(6,9)(7,14)(8,20)(10,19)(12,28)(13,27)(15,29)(17,26)(18,30)(22,24)(23,31)(25,32);(1,2)(3,12,13,7)(4,8)(5,10)(6,9)(11,14,27,28)(15,26,22,23)(16,20)(17,29,31,24)(18,25)(19,21)(30,32);(1,3)(2,7)(4,17)(5,22)(6,13)(8,24)(9,12)(10,26)(11,16)(14,20)(15,19)(18,31)(21,23)(25,29)(27,30)(28,32);(4,18)(8,25)(11,27)(14,28)(16,30)(17,31)(20,32)(24,29)Polycyclic presentation:
2^9|b^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fh,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72214675 |
| maximal normal subgroup of | #72214675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407156);