Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,13,11)(12,15,16,14)(17,18)(19,21)(20,23)(22,24);(5,6,8,7)(10,11)(14,15)(18,19)(22,23);(17,18,21,19)(20,22,24,23);(1,3,4,2)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,23,24,22);(9,12,13,16)(10,14,11,15)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df152 conjugacy classes — view the full character table.
| maximal subgroup of | #72261236, #72261462, #72261550, … |
| maximal normal subgroup of | #72261236, #72261462, #72261550, … |
| maximal quotient of | #72298404, C2×#513833 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421029);