Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #253411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #47881 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1019 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)(3,11)(4,15)(8,22)(9,20)(10,12)(13,26)(14,29)(16,17)(19,23)(21,31)(24,32)(25,28)(27,30);(2,8,9,23)(3,11)(4,16)(7,19,20,22)(10,24,25,31)(12,21,28,32)(13,26)(14,30)(15,17)(27,29);(1,2,6,10)(3,12,11,7)(4,8,17,24)(5,9,18,25)(13,28,26,20)(14,19,27,21)(15,31,16,23)(22,30,32,29);(3,13)(7,20)(11,26)(12,28)(14,29)(19,22)(21,32)(27,30)Polycyclic presentation:
2^9|b^2=e,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338704, #72341210, #72341226 |
| maximal normal subgroup of | #72338704, #72341210, #72341226 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 160609);