Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186835 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,20)(3,12)(4,15)(8,22,24,32)(9,16,11,21)(13,28)(14,27)(17,30)(19,25,31,23)(26,29);(1,2)(3,8)(4,16)(5,11)(6,9)(7,17)(10,18)(12,23)(13,22)(14,25)(15,20)(19,26)(21,30)(24,27)(28,32)(29,31);(4,17)(7,16)(13,26)(15,30)(19,22)(20,21)(28,29)(31,32);(1,3,5,14)(2,8,11,25)(4,13,15,29)(6,12,18,27)(7,19,21,32)(9,23,10,24)(16,22,20,31)(17,26,30,28)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213843, #72213902, #72215564, … |
| maximal normal subgroup of | #72213843, #72213902, #72215564, … |
| maximal quotient of | C2×#186837, #72240302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94033);