Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,12,21,18)(11,16,13,17,22,20,24,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,14)(15,22)(17,19)(18,23)(21,24);(9,12)(10,15)(11,20)(13,23)(14,18)(16,22)(17,24)(19,21);(3,8)(5,7)(9,13)(10,17)(11,21)(12,23)(14,22)(15,24)(16,18)(19,20);(2,6)(5,7)(11,22)(13,24)(16,20)(17,23)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257152, #72311985, #72312641, … |
| maximal normal subgroup of | #72257152, #72311985, #72312641, … |
| maximal quotient of | #72264624, #72264630, #72264644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293275);