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×#546 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,15,14)(11,16,21,20)(12,17,19,22)(13,18,24,23);(5,6)(7,8)(9,11,15,21)(10,16,14,20)(12,13,19,24)(17,18,22,23);(10,14)(16,20)(17,22)(18,23);(9,12)(10,17)(11,13)(14,22)(15,19)(16,18)(20,23)(21,24);(9,13)(10,18)(11,19)(12,21)(14,23)(15,24)(16,22)(17,20);(1,3,4,2)(9,14,15,10)(11,20,21,16)(12,22,19,17)(13,23,24,18)Polycyclic presentation:
2^9|a^2=i,b^2=h,f^2=i,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
| maximal subgroup of | #72284065, #72284104, #72335282 |
| maximal normal subgroup of | #72284065, #72284104, #72335282 |
| maximal quotient of | #72335318, #72335675, C22⋊C4×#546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533067);