Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
23366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24);(1,2)(3,4)(5,6)(7,8)(9,11,14,20)(10,15,16,22)(12,13,19,21)(17,18,23,24);(10,16)(15,22)(17,23)(18,24);(9,12)(10,17)(11,13)(14,19)(15,18)(16,23)(20,21)(22,24);(9,13)(10,18)(11,19)(12,20)(14,21)(15,23)(16,24)(17,22);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
| maximal subgroup of | #72234474, #72234478, #72256632, … |
| maximal normal subgroup of | #72234474, #72234478, #72256632, … |
| maximal quotient of | D4×#2339, #72263787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532973);