Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(14,20)(15,22);(2,5)(6,8)(9,11,13,18)(10,14,16,20)(12,17,19,23)(15,21,22,24);(9,12)(10,15)(11,17)(13,19)(14,21)(16,22)(18,23)(20,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=i,e^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249888, #72257443, #72257536, … |
| maximal normal subgroup of | #72249888, #72257443, #72257536, … |
| maximal quotient of | #72257361, C4×#24260, C2×#6454043 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361239);