Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,17,14,18,22,24)(11,15,20,23,19,12,16,21);(1,2,3,4)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(6,8);(9,12,13,21,14,15,22,23)(10,16,17,11,18,20,24,19);(11,19)(12,15)(16,20)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,e^b=ei,e^d=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72307265, #72307298, #72307324, … |
| maximal normal subgroup of | #72307265, #72307298, #72307324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279973);