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 | C2×#3489 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10868 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)(6,8)(9,10)(11,14)(12,17)(13,18)(15,16)(19,21)(20,23)(22,24);(5,6)(7,8)(9,11,15,20)(10,14,16,23)(12,13,19,22)(17,18,21,24);(1,3,4,2)(10,16)(14,23)(17,21)(18,24);(9,12)(10,17)(11,13)(14,18)(15,19)(16,21)(20,22)(23,24);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,24)(17,23);(9,14,15,23)(10,11,16,20)(12,18,19,24)(13,21,22,17)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,f^2=h,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, 7533037);