Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1929 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,24)(19,22,20,23);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,20)(18,22,24,23);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,23)(19,20)(22,24);(2,6)(7,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282132, #72282375, #72305296 |
| maximal normal subgroup of | #72282132, #72282375, #72305296 |
| maximal quotient of | #72305711, #72305718, #72315835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7474162);