Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2424 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,20,21,23);(2,3)(9,11,13,16)(10,14,12,15)(17,19,22,21)(18,20,24,23);(9,10,13,12)(11,15,16,14)(17,20,22,23)(18,19,24,21);(2,3)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,18)(19,20)(21,23)(22,24);(1,3,4,2)(9,11)(10,15)(12,14)(13,16)(17,21,22,19)(18,23,24,20)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72281698, #72281722, #72329569, … |
| maximal normal subgroup of | #72281698, #72281722, #72329569, … |
| maximal quotient of | #72331595, #72332973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 685075);