Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 1048576 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(18,21)(22,23);(2,4)(9,11,14,13)(10,15,16,12)(18,21)(22,23);(2,4)(6,7)(10,16)(11,13)(17,18)(19,23)(20,21)(22,24);(9,12)(10,11)(13,16)(14,15);(1,2)(3,4)(5,7,8,6)(9,13,14,11)(10,12,16,15)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,e^2=fg,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 | #72264709, #72329469, #72329483, … |
| maximal normal subgroup of | #72264709, #72329469, #72329483, … |
| maximal quotient of | #72329992, #72330537, #72330646 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214080);