Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460462 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2499 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16698 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,19)(4,16,18,20)(8,22,11,25)(12,30,31,17)(24,29)(26,32);(1,2,6,10)(3,11)(4,9,18,28)(5,19,21,7)(8,15)(12,29)(13,22)(14,25)(16,23,20,27)(17,26)(24,31)(30,32);(1,3)(2,8)(4,17,18,30)(5,13)(6,15)(7,22)(9,26,28,32)(10,11)(12,16,31,20)(14,21)(19,25)(23,24,27,29);(1,4)(2,9)(3,12)(5,20)(6,18)(7,23)(8,24)(10,28)(11,29)(13,30)(14,17)(15,31)(16,21)(19,27)(22,32)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377951);