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×#3500 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11158 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)(11,17)(12,18)(13,19)(14,16)(15,21)(20,23)(22,24);(5,6,7,8)(9,11)(10,15)(12,20)(13,18)(14,21)(16,17)(19,22)(23,24);(1,3,4,2)(10,16)(13,23)(15,17)(18,24);(9,11)(10,17)(12,19)(13,18)(14,21)(15,16)(20,22)(23,24);(9,12)(10,18)(11,20)(13,15)(14,22)(16,24)(17,23)(19,21);(9,13,14,23)(10,19,16,20)(11,18,21,24)(12,15,22,17)Polycyclic presentation:
2^9|a^2=i,b^2=gh,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.
| maximal subgroup of | #72284735, #72284895, #72284953, … |
| maximal normal subgroup of | #72284735, #72284895, #72284953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533038);