Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2308 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2308 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4303 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)(3,14,17,8)(4,18,21,10)(5,11)(6,13)(7,12)(9,16,25,22)(15,23)(19,27,20,26)(24,29)(28,30)(31,32);(1,3)(2,8)(4,15)(5,16)(6,22)(7,17)(9,11)(10,23)(12,14)(13,25)(18,28)(19,29)(20,32)(21,30)(24,26)(27,31);(1,3)(2,9)(4,15)(5,16)(6,22)(7,17)(8,13)(10,24)(11,14)(12,25)(18,31)(19,29)(20,32)(21,30)(23,27)(26,28);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,23)(9,24)(11,26)(12,18)(13,27)(14,28)(16,29)(17,30)(22,32)(25,31);(1,5,7,6)(2,11,12,13)(3,16,17,22)(4,19,21,20)(8,9,14,25)(10,26,18,27)(15,29,30,32)(23,24,28,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72258084, #72261065, #72261154, … |
| maximal normal subgroup of | #72258084, #72261065, #72261154, … |
| maximal quotient of | #72262952, #72302970, #72303077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281399);