Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,8,7)(13,16);(5,7,8,6)(9,10)(11,13)(12,14)(15,16);(1,2,3,4);(9,11)(10,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,b^a=be,c^a=cf,d^a=dg,g^b=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#32926, #72281637, #72281648, … |
| maximal normal subgroup of | C4×#32926, #72281637, #72281648, … |
| maximal quotient of | #72282477, #72282608, #72282963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300096);