Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #492504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17);(2,5)(3,8)(6,7);(9,11)(10,15)(12,22)(13,20)(14,23)(16,21)(17,19)Polycyclic presentation:
2^9|b^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=de,e^b=eg,g^b=gi,f^c=fh,e^d=eg,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | D8×#509, #72342593, #72342600, … |
| maximal normal subgroup of | D8×#509, #72342593, #72342600, … |
| maximal quotient of | #72226892, #72226899, #72226945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399701);