Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10,14,15,13,16,12,18)(11,17,21,22,20,23,19,24);(5,7,8,6)(10,15)(11,19)(12,14)(16,18)(20,21)(22,24);(1,2,3,4)(10,16)(15,18)(17,23)(22,24);(9,11,12,19,13,20,14,21)(10,17,18,24,16,23,15,22);(10,16)(11,20)(15,18)(19,21)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72282815, #72283321, #72283418, … |
| maximal normal subgroup of | #72282815, #72283321, #72283418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289991);