Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=f,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fg,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231802, #72231818, #72231827, … |
| maximal normal subgroup of | #72231802, #72231818, #72231827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387852);