Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 | #62759 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3289 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)(9,10,13,17,14,18,12,22)(11,15,21,24,19,23,20,16);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,22)(17,24)(18,23);(9,11)(10,16,18,24)(12,20)(13,21)(14,19)(15,22,23,17);(3,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,22)(17,24)(18,23);(2,6)(5,7)(11,19)(15,23)(16,24)(20,21)Polycyclic presentation:
2^9|a^2=h,c^2=gh,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257161, #72319562, #72319820 |
| maximal normal subgroup of | #72257161, #72319562, #72319820 |
| maximal quotient of | #72264630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300718);