Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2274 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)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18,19,23,20,22,24,21);(9,10,13,14)(11,16,12,15)(18,21)(19,24)(22,23);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,22)(21,23);(9,12,13,11)(10,16,14,15);(1,3,4,7)(2,6,5,8)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224781, #72224785, #72283031, … |
| maximal normal subgroup of | #72224781, #72224785, #72283031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375815);