Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C22×#120 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,8,5,6,3,7)(9,10)(11,15,16,13)(12,14);(10,13)(14,15);(2,6)(7,8);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,f^b=fg,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72230994, #72231038, #72231046, … |
| maximal normal subgroup of | #72230994, #72231038, #72231046, … |
| maximal quotient of | C2×#480376, #72262071, #72262074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387572);