Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,15)(14,16)(19,23)(21,24);(5,6)(7,8)(9,11,14,12)(10,15,16,13)(17,18,20,22)(19,21,23,24);(2,4)(6,8)(10,16)(13,15)(17,18)(19,21)(20,22)(23,24);(9,12,14,11)(10,15,16,13)(17,19)(18,21)(20,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,13,14,15)(10,11,16,12)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72264361, #72264438, #72264460, … |
| maximal normal subgroup of | #72264361, #72264438, #72264460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213734);