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) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#550 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(5,6,7,8)(9,10,15,14)(11,16,12,17)(13,18,22,21)(19,23,20,24);(5,6)(7,8)(9,11,15,12)(10,16,14,17)(13,19,22,20)(18,23,21,24);(10,14)(16,17)(18,21)(23,24);(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24);(9,13,15,22)(10,18,14,21)(11,19,12,20)(16,23,17,24);(1,3,4,2)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533108);