Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9467 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,21,25,22,24,20,30)(19,26,29,31,28,32,27,23);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23,24,31)(20,27)(21,29)(22,28)(25,26,30,32);(18,24)(20,21)(23,31)(27,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,f^b=fi,e^c=ei,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238934, #72238952, #72257890, … |
| maximal normal subgroup of | #72238934, #72238952, #72257890, … |
| maximal quotient of | #72264435 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108725);