Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15267 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2314 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)(3,12)(4,16,19,9)(5,20)(6,10)(7,11)(8,14)(13,26,30,29)(15,24)(17,23,31,32)(18,27,21,28)(22,25);(1,3,5,14,7,15,6,22)(2,8,10,12,11,25,20,24)(4,17,18,30,19,31,21,13)(9,26,27,32,16,29,28,23);(1,2)(3,12)(4,9)(5,10)(6,20)(7,11)(8,22)(13,29)(14,25)(15,24)(16,19)(17,23)(18,27)(21,28)(26,30)(31,32);(1,4)(2,9)(3,13)(5,21)(6,18)(7,19)(8,23)(10,28)(11,16)(12,29)(14,31)(15,30)(17,22)(20,27)(24,26)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72215953, #72215974, … |
| maximal normal subgroup of | #72213603, #72215953, #72215974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326242);