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 | #15124 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4782 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)(9,10,13,12)(11,15,16,14)(18,21)(22,23);(5,6)(7,8)(10,12)(14,15)(17,18,20,21)(19,23,24,22);(2,4)(6,8)(17,18)(19,23)(20,21)(22,24);(9,11,13,16)(10,14,12,15);(1,2)(3,4)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,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 | #72264545, #72264568, #72264617, … |
| maximal normal subgroup of | #72264545, #72264568, #72264617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214727);