Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622451 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#99 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,9)(5,16,17,7)(6,10)(11,20)(12,21)(13,27,25,18)(14,23)(15,22,28,19)(24,30)(26,31,32,29);(2,7)(4,14)(8,18)(9,22)(10,16)(12,24)(15,28)(19,23)(20,27)(21,31)(26,32)(29,30);(1,3)(2,8)(4,12)(5,13)(6,11)(7,18)(9,21)(10,20)(14,24)(15,26)(16,27)(17,25)(19,29)(22,31)(23,30)(28,32);(3,11)(8,20)(12,24)(13,25)(18,27)(21,30)(26,32)(29,31);(1,4,6,14)(2,9,10,23)(3,12,11,24)(5,15,17,28)(7,19,16,22)(8,21,20,30)(13,26,25,32)(18,29,27,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,g^b=gh,d^c=di,g^f=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72302917, #72302953, #72303048, … |
| maximal normal subgroup of | #72302917, #72302953, #72303048, … |
| maximal quotient of | #72311264, #72311463, #72311478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529649);